diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzoxcq" "b/data_all_eng_slimpj/shuffled/split2/finalzzoxcq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzoxcq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThere is a body of literature on an ideal Bose gas in a uniform gravitational \nfield~\\cite{gersch57,widom68,baranov01,cavalcanti02,uncu13}.\n The gas may be contained in an external potential, or a large box, and \nsubjected to a uniform gravitational potential. Using the semiclassical \napproximation, its thermal properties have been calculated analytically \nin the grand canonical formalism~\\cite{liu09a,du12}. Our motivation for studying this \nsimple system is to check how closely does the semiclassical approximation follow \nthe results of the quantum calculation. The authors of Ref.~\\cite{cavalcanti02} claim that, contrary to previous wisdom, in three dimensions an ideal Bose gas in a uniform gravitational field does not undergo BEC at a finite temperature. They attribute that to the replacement of the discrete quantum energy spectrum with a smooth density of states.\nIn the semiclassical approximation, one replaces the discrete density of states by a smooth one, while treating the ground state exactly. In the quantum calculation, on the other hand, the exact discrete energy levels of the system are calculated to compute the grand canonical ensemble (GCE) \nand the resulting thermal properties. In realistic statistical mechanics problems, one generally follows the semiclassical route. For the system at hand, the quantum \ncalculation is done for a finite number of particles. We find that as the number of particles is increased to larger and larger values, the quantum and semiclassical results become very close, even across the BEC critical temperature. \n\nIn this paper, we pay special \nattention to the calculation of the isothermal compressibility of the Bose gas. \nRecent experimental work on the isothermal compressibility across the Bose-Einstein condensation has been reported in Ref.~\\cite{poveda-cuevas15} for a harmonically trapped gas. The authors suggest that the isothermal compressibility around the critical pressure reveals a second-order nature of the phase transition. On the other hand predictions based on a number of different mean-field approximations~\\cite{olivares-quiroz10} do not lead to second-order phase transitions, and the isothermal compressibility does not diverge at criticality. In contrast to these authors, we discuss noninteracting systems only.\n It is well documented that in GCE, the isothermal compressibility diverges at the critical temperature $T_c$ in the absence of interparticle interactions~\\cite{\nlandau58}. \nIt is also known that even a weak \ninterparticle interaction removes this divergence~\\cite{bhaduri02}. In the present \nproblem, however, gravitation is introduced as a one-body ramp potential, and it \nis not clear at the outset how it will affect the compressibility. \n\nWe find that the semiclassical calculation in three dimensions of the ideal Bose gas with uniform gravity is equivalent to the analysis of a five-dimensional ideal Bose gas without gravity. We use this novel approach to obtain results for the specific heat and isothermal compressibility. \n The resulting compressibility is divergence-free and continuous across $T_c$. In the case of heat capacity, \n in the absence of the gravitational field, there is a discontinuity in its slope at \n$T_c$. Introducing gravitation, or, alternately five spatial dimensions, this \ndiscontinuity is in the heat capacity itself. \n\nThe calculations were performed by \ntaking a cylindrical container, as shown in Fig.~\\ref{fig:01}. \nIn the $zx$ plane, we take a \ncircular disc, which is the bottom of the cylinder at $y=0$. The atoms in the Bose \ngas are not allowed to take negative values of $y$. The gravitational field is along \nthe $y$ direction, and the potential is a ramp along the positive $y$ axis. \n\nThe plan of the paper is as follows. In Sec.~\\ref{sec:2}, the semiclassical calculation is \ndone using the phase space approach. It is established that one can describe the \nsystem under consideration in five spatial dimensions, but without the gravitation.\n The grand potential is calculated and the critical temperature $T_c$ is obtained. \nIn Sec.~\\ref{sec:3}, we give the results for isothermal compressibility and the \nheat capacity. In Sec.~\\ref{sec:4} a quantum calculation is done to show that BEC takes \nplace and the results agree with the semiclassical calculation. \n\n\\section{Three-dimensional gas in a uniform gravitational potential} \n\\label{sec:2}\nIn this section, we show that an ideal Bose gas in three spatial dimensions, \nsubjected to a uniform gravitational potential, may be looked upon as an \nideal five dimensional gravity-free gas. We then use the semiclassical method to \ncalculate the critical temperature of BEC. \n\\begin{figure}[!h]\n\\begin{center}\n\\begin{tikzpicture}[scale = 0.5,line width = 1.2pt]\n\\draw[->](0,0)-- (5,0) node [right]{$x$};\n\\draw[->](0,0)--(0,5) node [above]{$y$};\n\\draw[->](0,0)--(-3,-3) node [below,left]{$z$}; \n\\draw(0,0) node [left]{$O$};\n\\draw (0,0) ellipse (3cm and 1cm);\n\\draw(0,3) ellipse(3cm and 1cm);\n\\draw(3,0)--(3,3);\n\\draw(-3,0)--(-3,3);\n\\draw(3,1.5) node [right]{$L$};\n\\draw[->](2,4.5)--(1,3) ;\n\\draw (2,4.5) node [above,right]{Area $=A=\\pi a^2$}; \n\\end{tikzpicture}\n\\caption{The Bose gas is confined to a cylindrical box with the downward gravitational force parallel to the $y$ axis.}\n\\label{fig:01}\n\\end{center}\n\\end{figure}\n\nUsing the geometry of Fig.~\\ref{fig:01}, the single particle energy is given by\n\\begin{equation}\n\\epsilon (p,y)=\\frac {p^2}{2m}+mgy ,\n\\end{equation}\nwhere $m$ is the mass of each boson, and $g$ is the gravitational acceleration \non the earth's surface, and $p^2=(p_x^2+p_y^2+p_z^2)$. \n The grand potential is given by ($k_B=1$) \n\\begin{equation}\n\\Omega_b=T\\sum_n \\ln \\left (1-z\\exp(-\\epsilon_n \\beta)\\right)\n=-T\\sum_{l=1}^{\\infty} \\frac{(z)^l}{l} Z_1(l\\beta)~,\n\\label{omega}\n\\end{equation}\nwhere $\\beta=1\/T$, the fugacity $z=\\exp(\\beta\\mu)$, and \n$Z_1(l\\beta)$ is the one-body partition function in the variable $l\\beta$.\nIn the semiclassical approximation, $Z_1(\\beta)$ in the variable $\\beta$, is \ngiven by \n\\begin{equation}\nZ_1(\\beta)=\\frac{1} {h^3} \\int d^3p~ e^{\\textstyle -\\beta p^2\/2m} \\int d^2 r\n\\int_0^L dy ~e^{\\textstyle -\\beta mgy}.\n\\end{equation}\nThe two-dimensional spatial integral gives the area $A$ of the disc, yielding \n\\begin{equation}\nZ_1(\\beta)=\\frac{A}{\\lambda_T^3}\\frac{(1-\\exp{(-\\beta mgL)})} {\\beta mg}~.\n\\label{dime1}\n\\end{equation}\nNote that as $g\\rightarrow 0$, we recover the correct \n$Z_1(\\beta)=\\dfrac{V}{\\lambda_T^3}$, \nwhere $V=A L$ is the three-dimensional spatial volume. \nThe thermal wavelength $\\lambda_T$ (obtained from the $p$ integration) is given by \n\\begin{equation}\n\\lambda_T=\\sqrt{\\frac{2\\pi\\hbar^2}{m T}}~.\n\\end{equation}\nFor our present problem with \\textit{nonzero} $g$ and low temperatures, we impose the condition that $k_B T \\ll mgL$, i.e. $\\beta mgL \\gg 1$. Under this condition, Eq.~(\\ref{dime1}) reduces to \n\\begin{equation}\\label{eq:6}\nZ_1(\\beta)=\\frac{A}{\\lambda_T^3}\\frac{1}{\\beta m g}~.\n\\end{equation} \nEquation~(\\ref{eq:6}) could be rewritten as an ideal five-dimensional partition function (without gravity)\n\\begin{equation}\n\\tilde{Z}_1(\\beta)=\\frac{V_5}{\\lambda_T^5}\n\\label{new}\n\\end{equation}\nwhere \n\\begin{equation}\nV_5=\\frac{2\\pi \\hbar^2 A}{m^2g}\n\\label{dime2}\n\\end{equation}\nhas the dimension of (length)$^5$. We write $V_5=(A\\cdot V_3)$, where $V_3$ is a \nhypothetical 3-volume. Taking $m$ to be that of a $Rb^{87}$ atom, we find \n$V_3$ to be exceedingly small, of the order of $10^{-18}$ cubic meter. \nThis $V_3$ is not to be confused with the large three-dimensional volume \n$V=AL$ in which the atoms are confined. \nIn the following, we shall calculate the thermal properties of this noninteracting gas \nof bosons in 5-spatial dimensions . \n \nSubstituting for $\\tilde{Z}_1(\\beta)$ from Eq.~(\\ref{new}) in Eq.~(\\ref{omega}), \nwe see that the grand potential may be written as \n\\begin{equation}\n\\Omega_b=-T \\frac{V_5}{\\lambda_T^5}\\sum_{l=1}^{\\infty}\\tilde{ b}_l z^l~\n=-T \\frac{V_5}{\\lambda_T^5}~ g_{7\/2}(z)~.\n\\label{cluster}\n\\end{equation}\nwhere $\\tilde{b}_l=1\/l^{7\/2}$ are the statistical ``cluster integrals''. \nIn standard notation, $\\displaystyle \\sum_{l=1}^{\\infty} z^l\/l^{7\/2}=g_{7\/2}(z) $~.\n\nIn the gas phase, \n\\begin{equation}\n\\bar{n}_5=\\frac{\\bar{N}}{V_5}=-\\frac{\\partial\\Omega_b}{\\partial\\mu}\n=\\frac{1}{\\lambda_T^5\n}\\sum_{l=1}\\tilde{b}_l z^l~=\\frac{1}{\\lambda_T^5} g_{5\/2}(z)~.\n\\label{deriv}\n\\end{equation}\nOne puts in the constraint that $\\bar{N}=N$ , and this makes $z$ a function \nof $T$. The sum on the RHS converges at $z=1$, so the above relation is valid \nonly for $T\\geq T_c$. For lower temperatures, the ground state starts having \nmacroscopic occupancies. The critical temperature is given by \n\\begin{equation}\n(\\bar{n}_5\\lambda_{T_c}^5)=\\zeta(5\/2)~,\n\\label{kya}\n\\end{equation}\nwhere $\\lambda_T$ is at $T=T_c$, and $\\zeta(5\/2)$ is the Riemann zeta function. \nIt is straight forward to deduce from Eq.~(\\ref{kya}) that the critical temperature \nis given by \n\\begin{equation}\nT_c=\\left(\\frac{\\bar{N}\\hbar^3g(2\\pi)^{3\/2}}{\\zeta(5\/2)A\\sqrt{m}}\\right)^{2\/5}~.\n\\label{cond}\n\\end{equation} \nThis agrees with the expression for $T_c$ as given by Du \\textit{et al.}~\\cite{du12}, that was \nobtained by the standard procedure in three spatial dimensions in the \npresence of the uniform gravitational field. Furthermore it follows that \n\\begin{equation}\\label{eq:13}\n\\dfrac{N_{\\epsilon=0}}{N} = 1 - \\left(\\dfrac{T}{T_c}\\right)^{5\/2} \\ \\ \\ \\mathrm{when} \\ \\ \\ T < T_c,\n\\end{equation}\nwhere $N_{\\epsilon=0}$ refers to the number of particles in the ground state.\nIn Sec.~\\ref{sec:4}, a fully \nquantum mechanical calculation is performed to demonstrate that BEC \ndoes take place at a finite temperature that is consistent with the \nsemiclassical result as given by Eq. (\\ref{cond}). \n \n\n\\section{Semiclassical isothermal compressibility and heat capacity}\n\\label{sec:3}\n\\subsection{Isothermal compressibility }\n\nQuite generally, the isothermal compressibility is defined, in any \ndimension, by \n\\begin{equation}\n\\kappa_T=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_T ~,\n\\end{equation}\nand is directly related to number fluctuation in GCE.\nNote that $\\kappa_T$ has different dimensionality in three and five \nspace dimensions. For this reason, we denote the compressibility in five dimensions \nby $\\tilde{\\kappa}_T$.\n\nOnce the grand potential $\\Omega_b$ has been calculated (see Eq.~(\\ref{cluster})),\nthe average particle number $\\bar{n}_5$ is obtained from \n$-\\dfrac{\\partial \\Omega_b}{\\partial\\mu}$ (see Eq.~(\\ref{deriv})), \nand its second derivative with respect to $\\mu$ is related to $\\kappa_T$ :\n\\begin{equation}\n-\\frac{\\partial^2 \\Omega_b}{\\partial\\mu^2}=\\frac{\\partial \\bar{N}}{\\partial\\mu}\n=V_5 \\bar{n}_5^2\\tilde{\\kappa}_T\n\\end{equation}\nNote that we fix $\\bar{N}=N$ using Eq.~(\\ref{deriv}), so that the number density of \nbosons remains a constant. \nA little algebra gives, for our five-dimensional ideal gas,\n\\begin{equation}\n\\tilde{\\kappa}_T=\\frac{1}{\\bar{n}_5^2}\\frac{\\beta}{\\lambda_T^5}{ \n\\sum_{l=1}^{\\infty}l^2\\tilde{b}_lz^l} = \\dfrac{1}{\\bar{n}_5k_BT}\\dfrac{g_{3\/2}(z)}{g_{5\/2}(z)},\n\\end{equation} \nwhere $\\tilde{b}_l=1\/l^{7\/2}$. Note that $\\tilde{\\kappa}_T$ is finite and continuous across \nthe critical temperature (with $z=1$). At $T_c$ the compressibility is finite, i.e., $\\bar{n}_5k_BT_c\\tilde{\\kappa}_T=1.9474$. \nThis is in contrast to the ideal gas in three spatial dimensions discussed in Ref.~\\cite{\npathria11}. In that case, the \ncompressibility is given as\n\\begin{equation}\\label{eq:16}\n\\kappa_T = \\dfrac{1}{\\bar{n}_3k_BT}\\dfrac{g_{1\/2}(z)}{g_{3\/2}(z)},\n\\end{equation}\nwhere $\\bar{n}_3=\\dfrac{N}{V} $ is the number density in three dimensions. \nThe compressibility diverges at $T_c$. In the presence of uniform gravitation, \nwe find that the above expression is modified to \n\\begin{equation}\\label{eq:16}\n\\kappa_T = \\dfrac{1}{\\bar{n}_3k_BT}\\dfrac{g_{3\/2}(z)}{g_{5\/2}(z)},\n\\end{equation}\nwhich is not divergent at $T_c$. Note that this expression is valid when $T\\geq T_c$. \n\nIn order to calculate the compressibility below as well as above $T_c$ we use a modification of Eq.~(\\ref{deriv}) that includes the effect of the ground state in order to obtain $z$ as a function of $T$, i.e.,\n\\begin{equation}\\label{eq:n3}\n\\bar{n}_5 = \\dfrac{1}{\\lambda_T^5}g_{5\/2}(z) + \\dfrac{1}{V_5}\\dfrac{z}{1-z}.\n\\end{equation} \nThe isothermal compressibility is then\n\\begin{equation}\n\\label{eq:n20}\n\\tilde{\\kappa}_T = \\dfrac{1}{\\bar{n}_5k_BT}\\dfrac{V_5g_{3\/2}(z)+ \\lambda_T^5\\dfrac{z}{(1-z)^2}}{V_5g_{5\/2}(z) + {\\lambda_T^5}\\dfrac{z}{(1-z)}},\n\\end{equation}\nwhich diverges at $T=0$.\nWe use the corresponding equation in three dimensions, with $\\kappa_T$ replacing $\\tilde{\\kappa}_T$, \n$\\bar{n}_3$ replacing $\\bar{n}_5$, and $V$ replacing $V_5$ in Eq.~(\\ref{eq:n20}), to compare with the quantum results in\nthree dimensions in the presence of uniform gravity.\n As we shall see, this procedure gives good agreement with the quantum calculations for both the heat capacity and the compressibility as shown in Fig.~\\ref{fig:06} in Sec.~\\ref{sec:4} of this paper.\n\n\\subsection{Heat capacity}\nTo calculate the heat capacity, we need to calculate the average energy $\\bar{E}$, \nwhich is $-\\dfrac{5}{2}\\Omega_b$, where $\\Omega_b$ is defined in Eq.~(\\ref{cluster}), \nand the average number of bosons $\\bar{N}$, defined in \nEq.~(\\ref{deriv}).\nThe energy is differentiated with respect to $T$, with the constraint that \n$\\bar{N}=N$, where $N$ is a constant. This gives the condition that \n$\\dfrac{d\\bar{N}}{dT}=0$, implying that the fugacity $z$ is dependent on $T$. \nThe derivation will not be given here since our result coincides with Du \\textit{et al.}~\\cite{du12}. We \ngive the final result below, so that this semiclassical result may be plotted \nnumerically, and compared with the quantum calculation in the next section. \n\\begin{equation}\\label{heat}\n\\def2.{2.}\n\\dfrac{C_V}{N} = \\left\\{ \\begin{array}{ll}\n\\dfrac{35}{4}\\dfrac{g_{7\/2}(z)}{g_{5\/2}(z)}-\n \\dfrac{25}{4}\\dfrac{g_{5\/2}(z)}{g_{3\/2}(z)},~~ &T>T_c \\\\\n \\dfrac{35}{4}\\dfrac{\\zeta(7\/2)}{\\zeta(5\/2)}\\left(\\dfrac{T}{T_c}\\right)^{5\/2}, &T20\\relax\n \\setcounter{#1}{0}%\n \\fi%\n}\n\\providecommand{\\footnoteref}{}\n\\renewcommand{\\footnoteref}[1]{\\protected@xdef\\@thefnmark{\\ref{#1}}\\@footnotemark}\n\\let\\@old@footnotetext\\footnotetext\n\\def\\footnotetext[#1]#2{%\n \\incrftnotectr{footnote}%\n \\@old@footnotetext[\\value{footnote}]{\\label{#1}#2}%\n}\n\n\\def\\kopfzeileleer{\n \\lhead[]{}\n \\chead[]{}\n \\rhead[]{}\n \\lfoot[]{}\n \\cfoot[]{}\n \\rfoot[]{}\n}\n\\def\\kopfzeiledefault{\n \\lhead[]{}\n \\lhead[]{}\n \\chead[]{}\n \\rhead[]{}\n \\lfoot[]{}\n 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\\normalsize\n}\n\n\\providecommand{\\highlightTerm}{}\n\\renewcommand{\\highlightTerm}[1]{\\emph{#1}}\n\\providecommand{\\highlightForReview}{}\n\\renewcommand{\\highlightForReview}[1]{%\n \\bgroup\\relax%\n \\color{blue}\\relax%\n #1\\relax%\n \\egroup\\relax%\n}\n\n\n\n\\def\\@adminfootnotes{%\n \\let\\@makefnmark\\relax\n \\let\\@thefnmark\\relax\n \\ifx\\@empty\\@date\\else%\n \\@footnotetext{\\@setdate}%\n \\fi%\n \\ifx\\@empty\\@subjclass\\else%\n \\@footnotetext{\\@setsubjclass}%\n \\fi\n \\ifx\\@empty\\@keywords\\else%\n \\@footnotetext{\\@setkeywords}%\n \\fi\n \\ifx\\@empty\\thankses\\else%\n \\@footnotetext{\\def\\par{\\let\\par\\@par}\\@setthanks}%\n \\fi\n}\n\n\\def\\@settitle{\\Large\\bfseries\\scshape\\@title}\n\n\\def\\@maketitle{%\n \\normalfont\\normalsize\n \\@adminfootnotes\n \\@mkboth{\\@nx\\shortauthors}{\\@nx\\shorttitle}%\n \\global\\topskip42\\p@\\relax\n {\\centering\\@settitle}\n \\ifx\\@empty\\authors\\else{\\centering\\small\\@setauthors}\\fi\n \\ifx\\@empty\\@date\\else{\\vtop{\\centering\\small\\@date\\@@par}}\\fi\n \\ifx\\@empty\\@dedicatory%\n \\else%\n \\baselineskip\\p@\n \\vtop{\\centering{\\footnotesize\\itshape\\@dedicatory\\@@par}%\n \\global\\dimen@i\\prevdepth}\\prevdepth\\dimen@i%\n \\fi\n \\@setabstract\n \\normalsize\n \\if@titlepage\n \\newpage\n \\else\n \\dimen@34\\p@\\advance\\dimen@-\\baselineskip\n \\fi\n}\n\n\\def\\addresseshere{%\n \\bgroup\n \\setlength{\\parindent}{0pt}\n \\enddoc@text\n \\egroup\n \\let\\enddoc@text\\relax\n}\n\n\\makeatother\n\n\n\n\\begin{document}\n\\startdocumentlayoutoptions\n\n\\thispagestyle{plain}\n\n\n\n\\defAbstract{Abstract}\n\\begin{abstract}\n We consider characterisations of unitary dilations and approximations\n of irreversible classical dynamical systems on a Hilbert space.\n \n In the commutative case,\n building on the work in \\cite{Chung1962exp}, one can express well known approximants\n (\\exempli Hille- and Yosida-approximants)\n via expectations over certain stochastic processes.\n Using this, our first result characterises\n the simultaneous regular unitary dilatability of commuting families of $\\Cnought$-semigroups\n via the dilatability of such approximants as well as via regular polynomial bounds.\n This extends the results in \\cite{Dahya2023dilation} to the unbounded setting.\n \n We secondly consider characterisations of unitary and regular unitary dilations\n via continuous and discrete functional calculi.\n Applying these tools to a large class of classical dynamical systems,\n these two notions of dilation exactly characterise\n when a system admits unitary approximations\n under certain distinct notions of weak convergence.\n This establishes a sharp topological distinction between the two notions of unitary dilations.\n Our results are applicable to commutative systems as well as\n non-commutative systems satisfying the \\emph{canonical commutation relations} (CCR) in the Weyl form.\n\\end{abstract}\n\n\n\n\\subjclass[2020]{47A20, 47D06, 60G55}\n\\keywords{Semigroups of operators; dilations; approximations; point processes; functional calculus; group $C^{\\ast}$-algebras.}\n\\title[Characterisations of dilations via approximants, expectations, and functional calculi]{Characterisations of dilations via approximants, expectations, and functional calculi}\n\\author{Raj Dahya}\n\\email{raj.dahya@web.de}\n\\address{Fakult\\\"at f\\\"ur Mathematik und Informatik\\newline\nUniversit\\\"at Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany}\n\n\\maketitle\n\n\n\n\\setcounternach{section}{1}\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Introduction]{Introduction}\n\\label{sec:introduction:sig:article-stochastic-raj-dahya}\n\n\n\\noindent\nClassical dynamical systems on Hilbert or Banach spaces,\nwhich are in general irreversible,\ncan be studied in at least two natural ways\nin terms of more ideal systems:\n via approximations\n and\n via embeddings into (or: \\usesinglequotes{dilations} to) larger reversible systems.\nTowards the former, see \\exempli \\cite{Hillephillips1957faAndSg,Butzer1967semiGrApproximationsBook,Chung1962exp,Krol2009}.\nThe study of the latter was in part inspired by a result from Halmos \\cite{Halmos1950dilation},\nand properly initiated by Sz.-Nagy and Foias in \\cite{Nagy1953,Nagy1970}\nwith their work on unitary (power) dilations of contractions\nand of $1$-parameter contractive $\\Cnought$-semigroups\nover Hilbert spaces.\nRemaining in the commutative setting,\nvarious results have been achieved for systems consisting of multiple operators\nas well as multi-parameter $\\Cnought$-semigroups\n(see \\exempli\n \\cite{Ando1963pairContractions,Slocinski1974,Slocinski1982,Ptak1985,LeMerdy1996DilMultiParam,Shamovich2017dilationsMultiParam}).\nFor a good overview, see \\exempli \\cite{Averson2010DilationOverview,Shalit2021DilationBook}.\n\nIn the \\textbf{first part} of this paper\n(\\S{}\\ref{sec:stochastic:sig:article-stochastic-raj-dahya}--\\ref{sec:first-results:sig:article-stochastic-raj-dahya}),\nwe shall first consider commutative systems of $\\Cnought$-semigroups.\nNote that a commuting family, $\\{U_{i}\\}_{i=1}^{d}$,\nof unitary $\\Cnought$-semigroups on a Hilbert space $\\HilbertRaum$\ncan be uniquely extended to an \\topSOT-continuous unitary representation,\n$U$, of $(\\reals^{d},+,\\zerovector)$ on $\\HilbertRaum$\ndefined via\n $\n U(\\mathbf{t})\n \\colonequals\n (\n \\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{t}^{-})}}\n U_{i}(t_{i}^{-})\n )^{\\ast}\n (\n \\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{t}^{+})}}\n U_{i}(t_{i}^{+})\n )\n $\nfor all $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\reals^{d}$,\nwhere\n $t^{-}=\\max\\{-t,\\,0\\}$\n and\n $t^{+}=\\max\\{t,\\,0\\}$\nfor $t\\in\\reals$\n(\\cf Stone's theorem \\cite[Theorem~I.4.7]{Goldstein1985semigroups}).\nBearing this in mind, a commuting family,\n $\\{T_{i}\\}_{i=1}^{d}$,\nof $\\Cnought$-semigroups on $\\HilbertRaum$\nis said to have a\n \\highlightTerm{simultaneous regular unitary dilation}\nif\n\n \\vspace{-3\\parskip}\n $$\n \\Big(\n \\prod_{i=1}^{d}T(t_{i}^{-})\n \\Big)^{\\ast}\n \\Big(\n \\prod_{i=1}^{d}T(t_{i}^{+})\n \\Big)\n =\n r^{\\ast}\\,U(\\mathbf{t})\\,r\n $$\n\n\\noindent\nholds for all $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\reals^{d}$,\nfor\n some $\\topSOT$-continuous unitary representation $U$ of $(\\reals^{d},+,0)$ on a Hilbert space $\\HilbertRaum_{1}$\n and\n some isometry ${r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}}$\n(and in this case we shall refer to the data\n $(\\HilbertRaum_{1},U,r)$\nas the simultaneous regular unitary dilation).\nA \\highlightTerm{simultaneous unitary dilation}\nis defined by the above condition restricted to $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n\nFor the cases $d\\in\\{1,2\\}$ it was proved in\n \\cite[Theorem~I.8.1]{Nagy1970},\n \\cite{Slocinski1974},\n \\cite[Theorem~2]{Slocinski1982},\n and\n \\cite[Theorem~2.3]{Ptak1985},\nthat all contractive commuting families $\\{T_{i}\\}_{i=1}^{d}$\nhave a simultaneous unitary dilation.\nIn the case of $d=1$, these are in fact regular unitary dilations.\nIn the general case of $d \\geq 1$,\nthe existence of simultaneous regular unitary dilations was fully characterised\nin \\cite[Theorem~3.2]{Ptak1985}\nvia a general condition which we may refer to as \\emph{Brehmer positivity}.\nLe~Merdy fully classified the existence of simultaneous unitary dilations\nas well as dilations \\emph{up to similarity}\nvia the complete boundedness of a certain functional calculus map\n(%\n see \\cite[Theorems~2.2~and~3.1]{LeMerdy1996DilMultiParam},\n which builds on \\cite[Corollaries~4.9~and~4.13]{Pisier2001bookCBmaps}%\n),\nand successfully applied the latter to commuting families of bounded analytic $\\Cnought$-semigroups.\nMoreover, Shamovich and Vinnivok provided\nsufficient embedding conditions on the generators\nfor the existence of simultaneous unitary dilations\n(see \\cite{Shamovich2017dilationsMultiParam}).\nRecent results contribute to this history by providing\ntwo further complete characterisations of the existence of simultaneous regular unitary dilations\nfor commuting families of $\\Cnought$-semigroups\nunder the assumption of bounded generators\n(see \\cite[Theorems~1.1~and~1.4]{Dahya2023dilation}).\nThe first characterisation was achieved\nvia the notion of \\emph{complete dissipativity}\n(see \\cite[Definition~2.8]{Dahya2023dilation}),\nwhich is a spectral-theoretic notion of a simple combinatorial expression involving the generators.\nThe second characterisation builds on the first\nand characterises the existence of regular unitary dilations\nvia \\emph{regular polynomial bounds}\n(see \\cite[Definition 6.2]{Dahya2023dilation}).\nFurthermore, analogue to \\cite{LeMerdy1996DilMultiParam},\nit was shown that all commuting families of $\\Cnought$-semigroups\nwith bounded generators have regular unitary dilations\nup to certain natural modification\n(see \\cite[Corollary~1.2]{Dahya2023dilation}).\n\nThe latter reference left open the question,\nwhether the characterisation via regular polynomial bounds\ncould be extended to the unbounded setting\n(see \\cite[Remark~6.6]{Dahya2023dilation}).\nMoreover, the characterisation via complete dissipativity involves a characterisation via approximants\nwhich raises the question, whether for certain \\emph{natural choices of approximants},\na commuting family of $\\Cnought$-semigroups has a simultaneous regular unitary dilation\nif and only if families of their approximants do.\nThe current paper shall answer both these questions positively\nin the general setting without the boundedness assumption\n(see \\S{}\\ref{sec:first-results:sig:article-stochastic-raj-dahya}).\n\nIn the \\textbf{second part} of this paper\n(\\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya}--\\ref{sec:second-results:sig:article-stochastic-raj-dahya}),\nwe consider classical dynamical systems more broadly described by\n\\emph{homomorphisms} defined over topological monoids.\nTo motivate this, observe that there is a natural correspondence between\ncommuting families, $\\{T_{i}\\}_{i=1}^{d}$, of\n(bounded\/contractive\/unitary) $\\Cnought$-semigroups\nand\n(bounded\/contractive\/unitary) \\topSOT-continuous homomorphisms,\n$T$, between $(\\realsNonNeg^{d},+,\\zerovector)$ and spaces of operators\n(\\cf \\cite[\\S{}1]{Dahya2023dilation})\nvia\n $T(\\mathbf{t}) = \\prod_{i=1}^{d}T_{i}(t_{i})$\n and\n $T_{i}(t) = T_{i}(t) = T(0,0,\\ldots,\\underset{i}{t},\\ldots,0)$\nfor $\\mathbf{t}\\in\\realsNonNeg^{d}$, $t\\in\\realsNonNeg$,\n$i\\in\\{1,2,\\ldots,d\\}$.\nIt is thus natural to consider homomorphisms defined over topological monoids.\nIf $G$ is a topological group and $M \\subseteq G$ is a submonoid\nand ${T:M\\to\\BoundedOps{\\HilbertRaum}}$ is an \\topSOT-continuous homomorphism from $M$\nto bounded operators on a Hilbert space $\\HilbertRaum$,\nwe define a \\highlightTerm{unitary dilation} of $T$\nto be a tuple $(\\HilbertRaum_{1},U,r)$,\nwhere\n $\\HilbertRaum_{1}$ is a Hilbert space,\n $U$ is an $\\topSOT$-continuous unitary representation $G$ on $\\HilbertRaum_{1}$,\n and\n ${r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}}$ is an isometry,\nsuch that\n\n \\vspace{-3\\parskip}\n $$\n T(x) = r^{\\ast}\\,U(x)\\,r\n $$\n\n\\noindent\nholds for all $x \\in M$.\nWe shall also consider monoids for which one can define\na \\emph{positivity structure},\nwhich consists of continuous maps ${\\cdot^{+},\\cdot^{-} \\colon G \\to M}$\n(see \\Cref{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}).\nUsing this concept, we say that $(\\HilbertRaum_{1},U,r)$\nis a \\highlightTerm{regular unitary dilation} of $T$,\nif\n\n \\vspace{-3\\parskip}\n $$\n (T(x^{-}))^{\\ast}T(x^{+})\n =\n r^{\\ast}\\,U(x)\\,r\n $$\n\n\\noindent\nholds for all $x \\in G$.\nConsidering $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$, $d\\in\\naturals$,\nby the above mentioned correspondence one can see that these concepts\nagree with the definitions of\nsimultaneous (regular) unitary dilations of commuting families.\n\nWe present in \\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya}\ncharacterisations\nof unitary and regular unitary dilations\nvia the means of \\emph{functional calculi},\ninspired by Sz.-Nagy and le~Merdy.\nThese tools enable us to characterise the existence of unitary approximations\nfor a broad class of classical systems\n(see \\S{}\\ref{sec:second-results:sig:article-stochastic-raj-dahya}).\n\nBefore proceeding,\nwe recall the afore mentioned results from the bounded setting,\ndefine our terminology\nand state the main results of this paper.\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation via complete dissipativity]{Characterisation via complete dissipativity}\n\\label{sec:introduction:dissipativity:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor a $\\Cnought$-semigroup, $T$, on a Hilbert space $\\HilbertRaum$ with generator $A$,\nif $T$ is contractive then it has a regular unitary dilation\n(\\cf \\cite[Theorem~I.8.1]{Nagy1970}).\nAnd clearly, the latter necessarily requires $T$ to be contractive.\nOn the other hand, by the Lumer-Phillips form of the Hille-Yosida theorem,\n$T$ is contractive if and only if $A$ is dissipative\n(see \\cite[Theorem~I.3.3]{Goldstein1985semigroups}).\nThus the dissipativity of the generator characterises\nthe regular unitary dilatability of a $\\Cnought$-semigroup.\n\nIn the setting of commuting families of semigroups,\ndissipativity can be generalised as follows:\nFor each $k\\in\\naturalsZero$,\nthe \\highlightTerm{$k$\\textsuperscript{th}-order dissipation operators}\nare defined by\n\n \\vspace{-3\\parskip}\n $$\n (-\\tfrac{1}{2})^{\\card{K}}\n \\sum_{\\isPartition{(C_{1},C_{2})}{K}}\n \\Big(\n \\prod_{i\\in C_{1}}\n A_{i}\n \\Big)^{\\ast}\n \\prod_{j\\in C_{2}}\n A_{j}\n $$\n\n\\noindent\nfor $K \\subseteq \\{1,2,\\ldots,d\\}$ with $\\card{K} = k$,\nwhere $\\isPartition{(C_{1},C_{2})}{K}$ denotes\nthat $C_{1},C_{2} \\subseteq K$ form a partition of $K$.\nWe say that the generators $\\{A_{i}\\}_{i=1}^{d}$ are \\highlightTerm{completely dissipative},\nif the dissipation operators of all finite orders are positive\n(\\cf \\cite[Definition~2.8]{Dahya2023dilation}).\nThis notion leads to a characterisation result obtained in\n \\cite{Dahya2023dilation},\nwhich for the purposes of this paper we restate as follows:\n\n\\begin{thm}[Characterisation via complete dissipativity]\n\\makelabel{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\HilbertRaum$ be a Hilbert space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$\n be a commuting family of $\\Cnought$-semigroups on $\\HilbertRaum$,\n with generators $\\{A_{i}\\}_{i=1}^{d}$.\n If the semigroups have bounded generators,\n then the following are equivalent:\n\n \\begin{kompaktenum}{\\bfseries (a)}\n \\item\\punktlabel{1}\n The family $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation.\n \\item\\punktlabel{2}\n The generators $\\{A_{i}\\}_{i=1}^{d}$ are completely dissipative.\n \\item\\punktlabel{3}\n There exists a net $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha \\in \\Lambda}$\n consisting of\n a commuting families of $\\Cnought$-semigroups on $\\HilbertRaum$,\n which each have simultaneous regular unitary dilations,\n such that\n $$\n \\sup_{\\mathbf{t} \\in L}\n \\normLong{\n \\Big(\n \\prod_{i=1}^{d}T^{(\\alpha)}_{i}(t_{i})\n -\n \\prod_{i=1}^{d}T_{i}(t_{i})\n \\Big)\n \\xi\n }\n \\underset{\\alpha}{\\longrightarrow} 0\n $$\n for all\n $\\xi \\in \\HilbertRaum$\n and\n compact $L \\subseteq \\realsNonNeg^{d}$.\n \\end{kompaktenum}\n\n \\noindent\n Furthermore, the notion of convergence in \\punktcref{3}\n can be replaced by pointwise \\topSOT-convergence.\n The implication\n \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n also holds without the boundedness assumption.\n\\end{thm}\n\nSee \\cite[Theorem~1.1 and Remark~4.4]{Dahya2023dilation} for a proof.\nNote that in \\punktcref{3} the choice of nets of approximants\nis artificially constructed in \\cite{Dahya2023dilation}.\nThus it is not immediate that\n \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\nholds for any given choice (even in the bounded setting).\nThis leads to a natural question (in the general setting),\nwhich motivated the research in the present paper:\n\n\\begin{qstn}\n\\makelabel{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}\n Let $\\{T_{i}\\}_{i=1}^{d}$ be a commuting family of\n $\\Cnought$-semigroups on a Hilbert space $\\HilbertRaum$.\n For which choices of approximants,\n $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha\\in\\Lambda}$,\n does it hold that\n the simultaneous regular unitary dilatability of $\\{T_{i}\\}_{i=1}^{d}$\n implies that of each family\n $\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d}$?\n\\end{qstn}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation via polynomial bounds]{Characterisation via polynomial bounds}\n\\label{sec:introduction:polynomial:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor commuting operators\n $\\{S_{i}\\}_{i=1}^{d} \\subseteq \\BoundedOps{\\HilbertRaum}$,\n\\highlightTerm{regular polynomial evaluation}\nis defined as the unique linear map\n\n \\vspace{-3\\parskip}\n $$\n \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}] \\ni p\n \\mapsto p(S_{1},S_{2},\\ldots,S_{d}) \\in \\BoundedOps{\\HilbertRaum}\n $$\n\n\\noindent\nsatisfying\n\n \\vspace{-3\\parskip}\n $$\n p(S_{1},S_{2},\\ldots,S_{d})\n = \\Big(\\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{n}^{-})}}S_{i}^{-n_{i}}\\Big)^{\\ast}\n \\Big(\\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{n}^{+})}}S_{i}^{n_{i}}\\Big)\n $$\n\n\\noindent\nfor all monomials of the form\n $p = \\prod_{i=1}^{d}X_{i}^{n_{i}}$\nwith $\\mathbf{n} \\in \\integers^{d}$\n(\\cf \\cite[Definition~6.1]{Dahya2023dilation}).\nWe say that $\\{T_{i}\\}_{i=1}^{d}$ satisfies\n\\highlightTerm{regular polynomial bounds}\nif\n\n \\vspace{-3\\parskip}\n $$\n \\norm{\n p(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))\n } \\leq \\sup_{\\boldsymbol{\\lambda} \\in \\Torus^{d}}\n \\abs{\n p(\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d})\n }\n $$\n\n\\noindent\nholds for all $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}$,\nwhere $\\Torus$ is the unit circle in the complex plane.\nUsing these notions, a second characterisation is obtained\nin \\cite{Dahya2023dilation},\nwhich for the purposes of this paper, we restate as follows:\n\n\\begin{thm}[Characterisation via polynomial bounds]\n\\makelabel{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\HilbertRaum$ be a Hilbert space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$\n be a commuting family of $\\Cnought$-semigroups on $\\HilbertRaum$\n with generators $\\{A_{i}\\}_{i=1}^{d}$.\n If the semigroups have bounded generators,\n then the following are equivalent:\n\n \\begin{kompaktenum}{\\bfseries (a)}\n \\item\\punktlabel{1}\n The family $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation.\n \\item\\punktlabel{2}\n The family $\\{T_{i}\\}_{i=1}^{d}$ satisfies regular polynomial bounds.\n \\item\\punktlabel{3}\n For each ${K \\subseteq \\{1,2,\\ldots,d\\}}$\n and all ${\\mathbf{t}=(t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}}$\n the operator\n $p_{K}(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))$\n is positive, where\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n p_{K}\n \\colonequals\n \\displaystyle\n \\sum_{\\mathclap{\n \\isPartition{(C_{1},C_{2})}{K}\n }}\n \\displaystyle\n \\prod_{i \\in C_{1}}\n (1 - X_{i}^{-1})\n \\cdot\n \\displaystyle\n \\prod_{j \\in C_{2}}\n (1 - X_{j}).\\\\\n \\end{eqnarray*}\n\n \\item\\punktlabel{4}\n The generators $\\{A_{i}\\}_{i=1}^{d}$ are completely dissipative.\n \\end{kompaktenum}\n\n \\noindent\n Furthermore, the implications\n \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\n hold without the boundedness assumption.\n\\end{thm}\n\nDue to the inclusion of the intermediate step \\punktcref{3},\nwe sketch the proof for the reader's convenience.\n\n \\begin{proof}[of \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}, sketch]\n The equivalence of\n \\punktcref{1}, \\punktcref{2}, and \\punktcref{4}\n is proved directly in \\cite[Theorem~1.4]{Dahya2023dilation} in reliance upon \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}.\n For \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}\n the assumption of bounded generators is not required\n (see \\cite[Remark~6.6]{Dahya2023dilation}).\n\n Towards \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}:\n Suppose (without the boundedness assumption!)\n that the family $\\{T_{i}\\}_{i=1}^{d}$ satisfies regular polynomial bounds.\n Let\n $K \\subseteq \\{1,2,\\ldots,d\\}$\n and\n $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}$\n be arbitrary.\n Using binomial expansions, one can see that the regular polynomial, $p_{K}$,\n can be expressed as\n $\n p_{K}\n = \\prod_{i=1}^{d}(2 - X_{i} - X_{i}^{-1})\n $,\n and thus\n $\n p_{K}(\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d})\n = \\prod_{i=1}^{d}(2 - \\lambda_{i} - \\lambda_{i}^{-1})\n = \\prod_{i=1}^{d}(2 - 2\\mathop{\\mathfrak{R}\\mathrm{e}}\\lambda_{i})\n \\in [0, 4^{d}]\n $\n for all $\\boldsymbol{\\lambda} = (\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d}) \\in \\Torus^{d}$.\n It follows that\n $\n \\sup_{\\boldsymbol{\\lambda}}\n \\abs{1 - \\alpha p_{K}(\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d})}\n \\leq 1\n $\n for sufficiently small $\\alpha\\in\\realsPos$.\n Since the family of semigroups satisfies regular polynomial bounds,\n it follows that\n $\n \\norm{\\onematrix - \\alpha p_{K}(T_{1}(t_{1}), T_{2}(t_{2}), \\ldots, T_{d}(t_{d}))}\n \\leq 1\n $\n for sufficiently small $\\alpha\\in\\realsPos$.\n As argued in the proof of \\cite[Theorem~1.4]{Dahya2023dilation}, this implies that\n $p_{K}(T_{1}(t_{1}), T_{2}(t_{2}), \\ldots, T_{d}(t_{d}))$\n is a positive operator.\n\n Finally, the implication \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}\n (under the assumption of bounded generators)\n is proved exactly as in \\cite[Theorem~1.4]{Dahya2023dilation}.\n By taking limits of these positive operators appropriately scaled,\n one obtains that the generators are completely dissipative.\n \\end{proof}\n\nThis result raises the question\n(\\cf \\cite[Remark~6.6]{Dahya2023dilation}),\nwhich along with \\Cref{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}\nalso motivated the research in the current paper.\n\n\\begin{qstn}\n\\makelabel{qstn:polynomial:sig:article-stochastic-raj-dahya}\n Does the equivalence of\n \\eqcref{it:1:thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n and\n \\eqcref{it:2:thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n in\n \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n continue to hold without the assumption of bounded generators?\n\\end{qstn}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation via expectation-approximants]{Characterisation via expectation-approximants}\n\\label{sec:introduction:results:sig:article-stochastic-raj-dahya}\n\n\\noindent\nWe shall demonstrate a further characterisation related\nto the above two results\nwithout the assumption of bounded generators.\nThe key idea is to make the implication\n \\eqcref{it:1:thm:classification:dissipativity:sig:article-stochastic-raj-dahya}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\eqcref{it:3:thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\n of\n \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\nwork by considering suitable canonical choices for approximants\n(\\cf \\Cref{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}).\n\nLet $T$ be an arbitrary $\\Cnought$-semigroup on a Banach space $\\BanachRaum$\nwith generator\n ${A \\colon \\opDomain{A} \\subseteq \\BanachRaum \\to \\BanachRaum}$.\nWe now consider two concrete nets of approximants\nof the form\n $(\n T^{(\\lambda)}\n = (\n e^{tA^{(\\lambda)}}\n )_{\\mathbf{t} \\in \\realsNonNeg^{d}}\n )_{\\lambda \\in I}$\nfor some $I \\subseteq \\realsPos$ directed by increasing values of $\\lambda$.\n\nIf $\\omega_{0}(T)\\in[-\\infty,\\:\\infty)$ is the \\highlightTerm{growth bound} for $T$\n(\\cf \\cite[Proposition~I.5.5 and Definition~I.5.6]{EngelNagel2000semigroupTextBook},\n \\cite[Lemma~I.2.12]{Goldstein1985semigroups}%\n),\nthe \\highlightTerm{$\\lambda$\\textsuperscript{th}-Yosida-approximant}\nis defined by\n $\n T^{(\\lambda)} = (e^{tA^{(\\lambda)}})_{t\\in\\realsNonNeg}\n $\nfor each $\\lambda \\in (\\omega_{0}(T),\\infty)$,\nwhere\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:yosida-approx-generator:sig:article-stochastic-raj-dahya}\n A^{(\\lambda)}\n \\colonequals\n \\lambda A \\opResolvent{A}{\\lambda}\n = \\lambda^{2} \\opResolvent{A}{\\lambda}\n - \\lambda\\onematrix\n \\end{eqnarray}\n\n\\noindent\nis a bounded operator.\nThe Yosida-approximants satisfy\n\n \\vspace{-3\\parskip}\n $$\n \\sup_{t \\in L}\\norm{(T^{(\\lambda)}(t) - T(t))\\xi} \\longrightarrow 0\n $$\n\n\\noindent\nas ${(\\omega_{0}(T),\\infty) \\ni \\lambda \\longrightarrow \\infty}$\nfor all\n $\\xi \\in \\BanachRaum$\n and\n compact $L \\subseteq \\realsNonNeg$.\nFurthermore, if $T$ is contractive, then each of the Yosida-approximants are contractive.\n(For a proof of these classical results, see \\exempli\n \\cite[Theorem~G.4.3]{HytNeervanMarkLutz2016bookVol2},\n \\cite[Theorem~II.3.5, pp.~73--74]{EngelNagel2000semigroupTextBook},\n \\cite[(12.3.4), p.~361]{Hillephillips1957faAndSg}.%\n)\n\nNow, if a family, $\\{T_{i}\\}_{i=1}^{d}$, of commuting $\\Cnought$-semigroups\nhas a simultaneous regular unitary dilation,\nthen in particular each of the $T_{i}$ must be contractive,\nand thus the family, $\\{T^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$,\nof Yosida-approximants consists of contractive $\\Cnought$-semigroups\nfor each $\\boldsymbol{\\lambda} = (\\lambda_{i})_{i=1}^{d} \\in \\realsPos^{d}$\n(\\cf the subsequent paragraph below (3.8) in \\cite{EngelNagel2000semigroupTextBook}).\nFurthermore, the commutativity of the $T_{i}$ implies the commutativity of the resolvents,%\n\\footnote{\n see \\exempli \\cite[Theorem~1]{Abdelaziz1983commutingSemigroups},\n where this is proved under slightly more general assumptions.\n One can also directly verify this\n by relying on the Laplace integral representation of resolvents.\n}\nwhich by \\eqref{eq:yosida-approx-generator:sig:article-stochastic-raj-dahya}\nimplies the commutativity of the generators\n $\\{A^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$,\nwhich in turn implies that\n $\\{T^{(\\lambda_{i})}\\}_{i=1}^{d}$\nis a commuting family.\n\nAnother appropriate approximation is constructed in \\emph{Hille's first exponential formula}.\nFor $\\lambda\\in\\realsPos$ call $T^{(\\lambda)} = (e^{tA^{(\\lambda)}})_{t\\in\\realsNonNeg}$\nthe \\highlightTerm{$\\lambda$\\textsuperscript{th}-Hille-approximant} for $T$,\nwhere\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:hille-approx-generator:sig:article-stochastic-raj-dahya}\n A^{(\\lambda)}\n \\colonequals\n \\lambda (T(\\tfrac{1}{\\lambda}) - \\onematrix).\n \\end{eqnarray}\n\n\\noindent\nis a bounded operator.\nThen again it holds that\n ${T^{(\\lambda)}(t) \\longrightarrow T(t)}$\nfor ${\\realsPos \\ni \\lambda \\longrightarrow \\infty}$\n \\wrt the \\topSOT-topology\n uniformly in $t$\n on compact subsets of $\\realsNonNeg$\n(see \\cite[\\S{}1.2 and Theorem~1.2.2]{Butzer1967semiGrApproximationsBook}).\nIt is easy to verify that\n $\\norm{T^{(\\lambda)}(t)} \\leq e^{-\\lambda t}e^{\\lambda t \\norm{T(\\tfrac{1}{\\lambda})}}$.\nThus the Hille-approximants of contractive $\\Cnought$-semigroups\nare themselves contractive $\\Cnought$-semigroups\nwith bounded generators.\nMoreover, if $\\{T_{i}\\}_{i=1}^{d}$ is a commuting family of (contractive) $\\Cnought$-semigroups,\nthen for each $\\boldsymbol{\\lambda} = (\\lambda_{i})_{i=1}^{d} \\in \\realsPos^{d}$\nby \\eqcref{eq:hille-approx-generator:sig:article-stochastic-raj-dahya}\nthe generators\n $\\{A^{(\\lambda_{i})}\\}_{i=1}^{d}$\nclearly commute,\nso that the family of Hille-approximants\n $\\{T^{(\\lambda_{i})}\\}_{i=1}^{d}$\nis a commuting family of (contractive) $\\Cnought$-semigroups.\n\nIt turns out that these classically defined approximants in semigroup theory\ncan be naturally generalised to a class of approximants,\nwhich we shall call \\highlightTerm{expectation-approximants}\n(see \\Cref{defn:expectation-approximants:sig:article-stochastic-raj-dahya} below).\nWe now state the first main result of this paper:\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=false]\n\\begin{thm}\n\\makelabel{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\HilbertRaum$ be a Hilbert space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$\n be a commuting family of contractive $\\Cnought$-semigroups on $\\HilbertRaum$.\n Further let\n $(T^{(\\alpha)}_{i})_{\\alpha \\in \\Lambda_{i}}$\n be a net of expectation-approximants for $T_{i}$\n (\\exempli Hille- and Yosida-approximants)\n for each $i\\in\\{1,2,\\ldots,d\\}$.\n Then the following are equivalent:\n\n \\begin{kompaktenum}{\\bfseries (a)}\n \\item\\punktlabel{1}\n The family $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation.\n \\item\\punktlabel{2}\n The family $\\{T_{i}\\}_{i=1}^{d}$ satisfies regular polynomial bounds.\n \\item\\punktlabel{3}\n For each ${K \\subseteq \\{1,2,\\ldots,d\\}}$\n and all ${\\mathbf{t}=(t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}}$\n the operator\n $p_{K}(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))$\n is positive, where\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n p_{K}(X_{1},X_{2},\\ldots,X_{d})\n \\colonequals\n \\displaystyle\n \\sum_{\\mathclap{\n \\isPartition{(C_{1},C_{2})}{K}\n }}\n \\displaystyle\n \\prod_{i \\in C_{1}}\n (1 - X_{i}^{-1})\n \\cdot\n \\displaystyle\n \\prod_{j \\in C_{2}}\n (1 - X_{j}).\\\\\n \\end{eqnarray*}\n\n \\item\\punktlabel{4}\n The family,\n $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$,\n of approximants\n has a simultaneous regular unitary dilation\n for each $\\boldsymbol{\\alpha} = (\\alpha_{i})_{i=1}^{d} \\in \\prod_{i=1}^{d}\\Lambda_{i}$.\n \\end{kompaktenum}\n\n \\@ifnextchar\\bgroup{\\nvraum@c}{\\nvraum@bes}{1}\n\n\\end{thm}\n\\end{schattierteboxdunn}\n\nBy \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\nwe shall be able to positively address\n \\Cref{%\n qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya,%\n qstn:polynomial:sig:article-stochastic-raj-dahya%\n }.\nNote that the main implication to prove is\n \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4},\nas the others have essentially been proved in\n\\cite{Dahya2023dilation}\n(\\cf the discussion in \\S{}\\ref{sec:introduction:dissipativity:sig:article-stochastic-raj-dahya}--\\ref{sec:introduction:polynomial:sig:article-stochastic-raj-dahya}).\nNow, in the case of bounded generators the spectral-theoretic concept of complete dissipativity\nwas crucial for the characterisations.\nIn the unbounded setting, it is unclear how to extend the notion\nof dissipation operators and thus of complete dissipativity.\nNonetheless, one possibility and its limitations shall be discussed\n(\\cf \\Cref{rem:complete-dissipativity:sig:article-stochastic-raj-dahya}).\nInstead, we make use of stochastic methods,\nbuilding on the results in \\cite{Chung1962exp}.\nUsing suitable stochastic processes\nand expectations computed strongly via Bochner-integrals,\nwhich we lay out in \\S{}\\ref{sec:stochastic:sig:article-stochastic-raj-dahya},\nwe show that expectation-approximants\ncan be expressed in terms of their original semigroups.\nThis provides a key ingredient to prove \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}.\nThis shall all be covered in\n \\S{}\\ref{sec:stochastic:sig:article-stochastic-raj-dahya}--\\ref{sec:first-results:sig:article-stochastic-raj-dahya}.\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Special conditions on topological monoids]{Special conditions on topological monoids}\n\\label{sec:introduction:monoids:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn the second part of this paper, we concentrate on classical dynamical systems\ndefined more broadly over topological monoids.\nWe are particularly interested in topological monoids, $M$,\nwhich occur as (closed or at least measurable) submonoids of (ideally locally compact) topological groups $G$.\n\n\\begin{e.g.}\n Simple examples of locally compact topological groups and closed submonoids thereof\n include any discrete group and submonoid thereof,\n \\exempli $(\\integers^{d},+,\\zerovector)$ and $\\naturalsZero^{d}\\subseteq\\integers^{d}$ for $d\\in\\naturalsPos$.\n We also consider\n $\\realsNonNeg^{d}$\n as a closed submonoid of the locally compact topological group $(\\reals^{d},+,\\zerovector)$.\n\\end{e.g.}\n\n\\begin{e.g.}\n As a non-discrete and non-commutative example,\n consider the Heisenberg group\n $\\Heisenberg_{d}$ of order $2d+1$, $d\\in\\naturalsPos$,\n which can be represented topologically as\n ${\\Heisenberg_{d}=\\reals^{d}\\times\\reals^{d}\\times\\reals}$\n and algebraically via the operation%\n \\footnote{\n There are various different presentations of the Heisenberg group in the literature\n (\\cf \\cite[\\S{}1]{Coburn1999artHeisneberg}, \\cite[\\S{}10.1]{Deitmar2014bookHarmonicAn}, \\cite[\\S{}6.7.4]{Folland2015bookHarmonicAnalysis}).\n We choose this particular form for convenience.\n }\n\n \\vspace{-3\\parskip}\n $$\n (\\mathbf{x},\\mathbf{p},E)\n (\\mathbf{x}^{\\prime},\\mathbf{p}^{\\prime},E^{\\prime})\n = \\Big(\n \\mathbf{x} + \\mathbf{x}^{\\prime},\n \\mathbf{p} + \\mathbf{p}^{\\prime},\n E + E^{\\prime}\n + \\frac{1}{2}(\n \\brkt{\\mathbf{p}}{\\mathbf{x}^{\\prime}}\n - \\brkt{\\mathbf{p}^{\\prime}}{\\mathbf{x}}\n )\n \\Big)\n $$\n\n \\noindent\n for\n $\n (\\mathbf{x},\\mathbf{p},E),\n (\\mathbf{x}^{\\prime},\\mathbf{p}^{\\prime},E^{\\prime})\n \\in \\Heisenberg_{d}\n $.\n The identity element of $\\Heisenberg_{d}$ is clearly $(\\zerovector,\\zerovector,0)$\n and inverse of $g\\in\\Heisenberg_{d}$ is given by $g^{-1}=-g$,\n if we view $g$ as a vector in $\\reals^{2d+1}$.\n One can readily verify that\n\n \\vspace{-3\\parskip}\n $$\n \\Heisenberg^{+}_{d}\n \\colonequals\n \\{\n (\\mathbf{x},\\mathbf{p},E)\n \\mid\n \\mathbf{x},\\mathbf{p}\\in\\realsNonNeg^{d},\n E\\in\\reals\n \\}\n \\subseteq \\Heisenberg_{d}\n $$\n\n \\noindent\n is a closed submonoid.\n As a further non-commutative example,\n consider for some antisymmetric matrix $C \\in M_{d}(\\reals)$\n the closed subgroup\n $\n \\Heisenberg_{d,C}\n \\colonequals\n \\{\n (\\mathbf{x},C\\mathbf{x},E)\n \\mid\n \\mathbf{x}\\in\\reals^{d},\n E\\in\\reals\n \\}\n \\subseteq \\Heisenberg_{d}\n $\n and the closed submonoid\n $\n \\Heisenberg^{+}_{d,C}\n \\colonequals\n \\{\n (\\mathbf{x},C\\mathbf{x},E)\n \\mid\n \\mathbf{x}\\in\\realsNonNeg^{d},\n E\\in\\reals\n \\}\n \\subseteq \\Heisenberg_{d,C}\n $.\n Without loss of generality, we can replace the above representations of\n $\\Heisenberg_{d,C}$ and $\\Heisenberg^{+}_{d,C}$\n by\n $\\reals^{d}\\times\\reals$ and $\\realsNonNeg^{d}\\times\\reals$\n respectively.\n Observe that the group operation satisfies\n\n \\vspace{-3\\parskip}\n $$\n (\\mathbf{x},E)\n (\\mathbf{x}^{\\prime},E)\n = \\Big(\n \\mathbf{x} + \\mathbf{x}^{\\prime},\n E + E^{\\prime}\n + \\frac{1}{2}(\n \\brkt{C\\mathbf{x}}{\\mathbf{x}^{\\prime}}\n - \\brkt{C\\mathbf{x}^{\\prime}}{\\mathbf{x}}\n )\n \\Big)\n = \\Big(\n \\mathbf{x} + \\mathbf{x}^{\\prime},\n E + E^{\\prime} + \\brkt{C\\mathbf{x}}{\\mathbf{x}^{\\prime}}\n \\Big)\n $$\n\n \\noindent\n for $(\\mathbf{x},E),(\\mathbf{x}^{\\prime},E^{\\prime})\\in\\Heisenberg_{d,C}$.\n One can thus think of the $i$\\textsuperscript{th} and $j$\\textsuperscript{th}\n $x$-co-ordinates of elements of $\\Heisenberg_{d,C}$\n as being \\emph{correlated} with $C_{ij}$ encoding this correlation.\n\\end{e.g.}\n\n\\begin{e.g.}[Non-commuting families, Weyl Form of CCR]\n\\makelabel{e.g.:non-commuting-family-heisenberg-d-C:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\HilbertRaum$ be a Hilbert space,\n and\n $C\\in M_{d}(\\reals)$ be an antisymmetric matrix,\n \\idest\n $C = D - D^{T}$, where $D \\in M_{d}(\\reals)$\n is a strictly upper triangular matrix.\n One use of classical dynamical systems defined over $\\Heisenberg^{+}_{d,C}$\n is as follows is as follows:\n \n Consider an arbitrary \\topSOT-continuous homomorphism,\n ${T \\colon \\Heisenberg^{+}_{d,C} \\to \\BoundedOps{\\HilbertRaum}}$.\n Set\n ${\n T_{i} \\colonequals (T(t\\mathbf{e}_{i},0))_{t\\in\\realsNonNeg}\n }$\n for each $i\\in\\{1,2,\\ldots,d\\}$,\n and\n ${\n U \\colonequals (T(\\zerovector, E))_{E\\in\\reals}\n }$,\n whereby each\n $\\mathbf{e}_{i} \\colonequals (0,0,\\ldots,\\underset{i}{1},\\ldots,0)$\n denotes the canonical $i$\\textsuperscript{th} basis vector of $\\reals^{d}$.\n Then one can readily verify that\n $\\{T_{i}\\}_{i=1}^{d}$ is a family of $\\Cnought$-semigroups on $\\HilbertRaum$\n and\n $U$ is an \\topSOT-continuous representation of $(\\reals^{d},+,\\zerovector)$ on $\\HilbertRaum$\n which commutes with each of the $T_{i}$.\n Furthermore the relations\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:commutation-relations:sig:article-stochastic-raj-dahya}\n T_{j}(t) T_{i}(s)\n &= &U(2stC_{ij}) T_{i}(s) T_{j}(t)\n \\end{eqnarray}\n\n \\noindent\n hold for all $i,j\\in\\{1,2,\\ldots,d\\}$,\n which is a slight generalisation of\n the semigroup version of the \\emph{canonical commutation relations} (CCR)\n in the Weyl form.\n Such systems are of interest in quantum mechanics\n (see \\exempli \\cite[\\S{}5.2.1.2]{Bratteli1997bookQSM}).\n \n Conversely, one may consider an arbitrary family,\n $\\{T_{i}\\}_{i=1}^{d}$,\n of $\\Cnought$-semigroups on $\\HilbertRaum$\n and an arbitrary \\topSOT-continuous representation,\n $U$,\n of $(\\reals^{d},+,\\zerovector)$ on $\\HilbertRaum$,\n which commutes with each of the $T_{i}$,\n and such that the relations in\n \\eqcref{eq:commutation-relations:sig:article-stochastic-raj-dahya} hold.%\n \\footnote{\n For example consider\n $\\HilbertRaum = L^{2}(\\realsNonNeg^{m})$, $m\\in\\naturals$.\n Let\n $\\mathbf{u}^{(i)}\\in\\realsNonNeg^{m}$\n and either\n $\\lambda\\in\\iunit\\reals$\n and\n $\\boldsymbol{\\alpha}^{(i)}\\in\\reals^{m}$,\n or\n $\\lambda\\in\\complex$ with $\\mathop{\\mathfrak{R}\\mathrm{e}}\\lambda < 0$\n and\n $\\boldsymbol{\\alpha}^{(i)}\\in\\realsNonNeg^{m}$\n for each $i\\in\\{1,2,\\ldots,d\\}$.\n Set\n $U \\colonequals (e^{\\lambda t}\\onematrix)_{t\\in\\reals}$\n and\n $\n (T_{i}(t)f)(\\mathbf{x})\n \\colonequals\n e^{\n \\lambda t\n \\brkt{\\boldsymbol{\\alpha}^{(i)}}{\\mathbf{x}}\n }\n f(\\mathbf{x} + t\\mathbf{u}^{(i)})\n $\n for\n $t\\in\\realsNonNeg$,\n $f\\in L^{2}(\\realsNonNeg^{m})$,\n $\\mathbf{x}\\in\\realsNonNeg^{m}$,\n $i\\in\\{1,2,\\ldots,d\\}$.\n Then \\eqcref{eq:commutation-relations:sig:article-stochastic-raj-dahya} holds with\n $\n C \\colonequals (\n \\frac{1}{2}(\n \\brkt{\\boldsymbol{\\alpha}^{(i)}}{\\mathbf{v}^{(j)}}\n -\n \\brkt{\\boldsymbol{\\alpha}^{(j)}}{\\mathbf{v}^{(i)}}\n )\n )_{i,j=1}^{d}\n \\in M_{d}(\\reals)\n $.\n }\n Then, defining\n ${T \\colon \\Heisenberg^{+}_{d,C} \\to \\BoundedOps{\\HilbertRaum}}$\n via\n\n \\vspace{-3\\parskip}\n $$\n T(\\mathbf{x},E)\n \\colonequals\n U(E + \\brkt{D\\mathbf{x}}{\\mathbf{x}})\n \\prod_{i=1}^{d}\n T_{i}(x_{i})\n $$\n\n \\noindent\n for each $\\mathbf{x}\\in\\realsNonNeg^{d}$, $E\\in\\reals$,\n one can verify that $T$ is an \\topSOT-continuous homomorphism.\n These two constructions constitute a correspondence\n (this is left as an exercise to the reader)\n between families satisfying\n \\eqcref{eq:commutation-relations:sig:article-stochastic-raj-dahya}\n and \\topSOT-continuous homomorphisms defined over\n $\\Heisenberg^{+}_{d,C}$.\n\\end{e.g.}\n\nFor submonoids of topological groups,\nthere are natural conditions which will aid us when investigating properties (\\viz dilation)\nof classical dynamical systems defined over them.\nThe following definition is due to Mueller \\cite[\\S{}2]{Mueller1965positiveInIdentity}:%\n\\footnote{\n \\cf also \\cite[Appendix~A]{Dahya2022complmetrproblem},\n where a similar, but stronger, property called \\emph{positivity in the identity} is defined.\n}\n\n\\begin{defn}\n\\makelabel{defn:monoids:e-jointedness:sig:article-stochastic-raj-dahya}\n Let $G$ be a topological group and $M \\subseteq G$ be a measurable submonoid.\n Say that $M$ is \\highlightTerm{$e$-joint}\n if $\\lambda_{G}(U \\cap M) > 0$\n for all open neighbourhoods $U \\subseteq G$ of the identity $e \\in G$.\n\\end{defn}\n\n\\begin{e.g.}\n\\makelabel{e.g.:monoids:e-joint:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $p\\in\\mathbb{P}$ be a prime number,\n $C\\in M_{d}(\\reals)$ be an antisymmetric matrix,\n $G_{i}$ be a topological group,\n and $M_{i} \\subseteq G_{i}$ an $e$-joint measurable submonoid\n for $i\\in\\{1,2,\\ldots,d\\}$.\n\n \\begin{kompaktenum}{(i)}\n \\item\\punktlabel{1}\n Consider $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$.\n For an open neighbourhood $U \\subseteq G$ of $\\zerovector$,\n there exists $\\varepsilon>0$, such that $U \\supseteq (-\\varepsilon,\\:\\varepsilon)^{d}$.\n Thus\n $M \\cap U \\supseteq (0,\\:\\varepsilon)^{d}$,\n which is a non-empty and thus non-null subset of $G$.\n Hence $M$ is an $e$-joint submonoid.\n \\item\\punktlabel{2}\n Consider\n $\n (G,M)\n = (\\Heisenberg_{d},\\Heisenberg^{+}_{d})\n = (\n \\reals^{d}\\times\\reals^{d}\\times\\reals,\n \\realsNonNeg^{d}\\times\\realsNonNeg^{d}\\times\\reals\n )\n $.\n As in \\punktcref{1},\n for any open neighbourhood $U \\subseteq G$ of $(\\zerovector,\\zerovector,0)$,\n one has\n $M \\cap U \\supseteq (0,\\:\\varepsilon)^{d} \\times (0,\\:\\varepsilon)^{d} \\times (-\\varepsilon,\\:\\varepsilon)$\n for some $\\varepsilon > 0$,\n which is a non-empty and thus non-null subset of $G$.\n Hence $M$ is an $e$-joint submonoid.\n \\item\n Similar to \\punktcref{2} one can show that\n $\\Heisenberg^{+}_{d,C}$ is a (closed) $e$-joint submonoid\n of $\\Heisenberg_{d,C}$.\n \\item\n The submonoid $\\integers_{p} \\mathbin{\\setminus} \\{0\\}$ of non-zero $p$-adic integers\n within the locally compact multiplicative group,\n $(\\rationals_{p} \\mathbin{\\setminus} \\{0\\},\\cdot,1)$,\n of non-zero $p$-adic numbers is clearly $e$-joint,\n since it is a clopen subset.\n \\item\n By simple computations with product measures\n one can readily verify that\n $\\prod_{i=1}^{d}M_{i}$ is a measurable $e$-joint submonoid in $\\prod_{i=1}^{d}G_{i}$\n (\\cf \\cite[Proposition~A.7]{Dahya2022complmetrproblem}).\n \\end{kompaktenum}\n\n \\@ifnextchar\\bgroup{\\nvraum@c}{\\nvraum@bes}{1}\n\n\\end{e.g.}\n\n\\begin{defn}\n\\makelabel{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n Let $(G,\\cdot,e)$ be a (not necessarly locally compact!) topological group\n and $M \\subseteq G$ a submonoid.\n If ${\\cdot^{+} \\colon G \\to M}$ is a continuous function which satisfies\n\n \\begin{kompaktenum}{(i)}\n \\item\\punktlabel{ax:identity}\n $e^{+} = e$ for the identity element $e \\in G$;\n \\item\\punktlabel{ax:idempotent}\n $x^{++} = x^{+}$ for all $x\\in G$,\n \\idest $\\cdot^{+}$ is idempotent;\n and\n \\item\\punktlabel{ax:representation}\n $(x^{-})^{-1}x^{+} = x$\n for all $x\\in G$,\n where $x^{-} \\colonequals (x^{-1})^{+}$,\n \\end{kompaktenum}\n\n \\noindent\n then we call $(G,M,\\cdot^{+})$\n a \\highlightTerm{positivity structure}.\n\\end{defn}\n\n\\begin{e.g.}\n\\makelabel{e.g.:monoids:positivity:sig:article-stochastic-raj-dahya}\n Let $d\\in\\naturals$ and $C \\in M_{d}(\\reals)$ be an antisymmetric matrix.\n The pairs $(G,M)$ of topological groups and submonoids:\n $(\\reals^{d},\\realsNonNeg^{d})$,\n $(\\Heisenberg_{d},\\Heisenberg^{+}_{d})$,\n $(\\Heisenberg_{d,C},\\Heisenberg^{+}_{d,C})$\n admit natural positivity structures.\n Furthermore, if\n $(G_{i},M_{i},\\cdot^{+_{i}})$\n are positivity structures,\n then the product\n $(\\prod_{i=1}^{d}G_{i},\\prod_{i=1}^{d}M_{i})$\n admits a positivity via the pointwise definition.\n The constructions are presented in\n \\Cref{table:examples:monoids:positivity:sig:article-stochastic-raj-dahya}\n and left to the reader to verify.\n\\end{e.g.}\n\n\\begin{table}[!htb]\n \\begin{tabular}[t]{|p{0.15\\textwidth}|p{0.15\\textwidth}|p{0.4\\textwidth}|}\n \\hline\n Group $G$\n &Submonoid $M$\n &Description of ${\\cdot^{+} \\colon G\\to M}$\\\\\n \\hline\n \\hline\n $\\prod_{i=1}^{d}G_{i}$\n &$\\prod_{i=1}^{d}M_{i}$\n &$\\mathbf{x} \\mapsto (x_{i}^{+_{i}})_{i=1}^{d}$\\\\\n $(\\reals,+,0)$\n &$\\realsNonNeg$\n &$t \\mapsto \\max\\{t,0\\}$\\\\\n $(\\reals^{d},+,\\zerovector)$\n &$\\realsNonNeg^{d}$\n &$\\mathbf{t} \\mapsto (t_{i}^{+})_{i=1}^{d}$\\\\\n $\\Heisenberg_{d}$\n &$\\Heisenberg^{+}_{d}$\n &$(\\mathbf{x},\\mathbf{p},E)\n \\mapsto\n (\n \\mathbf{x}^{+},\n \\mathbf{p}^{+},\n E^{+}\n )\n $\\\\\n $\\Heisenberg_{d,C}$\n &$\\Heisenberg^{+}_{d}$\n &$(\\mathbf{x},E)\n \\mapsto\n (\n \\mathbf{x}^{+},\n E^{+}\n )\n $\\\\\n \\hline\n \\end{tabular}\n \\caption{\n Examples of positivity structures $(G,M,\\cdot^{+})$.\n Here $d\\in\\naturals$, $C \\in M_{d}(\\reals)$ is an antisymmetric matrix,\n and $(G_{i},M_{i},\\cdot^{+_{i}})$\n are positivity structures for $i\\in\\{1,2,\\ldots,d\\}$.\n }\n \\label{table:examples:monoids:positivity:sig:article-stochastic-raj-dahya}\n\\end{table}\n\nNote that for the subset\n $G^{+} \\colonequals \\{x^{+}\\mid g\\in G\\} \\subseteq M$,\nit is neither required that $G^{+} = M$\nnor even that $G^{+}$ be closed under the group operation.\nIn the case of $\\reals^{d}$, this happens to be the case,\nbut in the case of the Heisenberg group,\nneither of these additional properties holds.\nWe now state some basic facts about positivity structures.\n\n\\begin{prop}\n Let $(G,M,\\cdot^{+})$ be a positivity structure,\n where $(G,\\cdot,e)$ is a topological group\n and $M \\subseteq G$ is a submonoid.\n Then\n $(G^{-})^{-1} \\cap G^{+} = (G^{+})^{-1} \\cap G^{+} = \\{e\\}$.\n\\end{prop}\n\n \\begin{proof}\n First observe that\n $(G^{-})^{-1} \\cap G^{+} = ((G^{-1})^{+})^{-1} \\cap G^{+} = (G^{+})^{-1} \\cap G^{+}$\n Thus it suffices to prove that $(G^{+})^{-1} \\cap G^{+} = \\{e\\}$.\n Since $e^{+} = e$, the $\\supseteq$-inclusion holds.\n Towards the $\\subseteq$-inclusion,\n let $x \\in (G^{+})^{-1} \\cap G^{+}$ be arbitrary.\n Then $x = y^{+} = (z^{+})^{-1}$\n for some $y, z \\in G$.\n By the idempotence axiom,\n one has that\n $\n x^{+}\n = y^{++}\n = y^{+}\n = x\n $\n and\n $\n x^{-}\n = (x^{-1})^{+}\n = z^{++}\n = z^{+}\n = x^{-1}\n $.\n By axiom \\eqcref{it:ax:representation:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya},\n it follows that\n $x = (x^{-})^{-1}(x^{+}) = x^{-1}x = e$.\n \\end{proof}\n\n\\begin{prop}\n\\makelabel{prop:positivity-structure-basic:sig:article-stochastic-raj-dahya}\n Let $(G,\\cdot,e)$ be a topological group\n and $M \\subseteq G$ a submonoid.\n Let ${\\cdot^{+} \\colon G \\to M}$\n be a continuous map satisfying\n axiom \\eqcref{it:ax:representation:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n of \\eqcref{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}.\n Then axiom\n \\eqcref{it:ax:idempotent:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n is equivalent to\n the condition that $x^{+-}=e$ for all $x\\in G$.\n\\end{prop}\n\n \\begin{proof}\n Let $x \\in G$ be arbitrary.\n Then by\n axiom \\eqcref{it:ax:representation:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n one has\n $x^{+} = (x^{+-})^{-1}x^{++}$.\n It follows that\n $x^{++} = x^{+}$\n if and only if\n $x^{+-} = e$.\n \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation of unitary approximations]{Characterisation of unitary approximations}\n\\label{sec:introduction:applications:sig:article-stochastic-raj-dahya}\n\n\\noindent\nLetting $M$ be a topological monoid and $\\HilbertRaum$ a Hilbert space,\na classical dynamical system modelled by a homomorphism,%\n\\footnote{\n \\idest $T(e) = \\onematrix$ and $T(xy) = T(x)T(y)$ for $x,y\\in M$.\n}\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$,\ncan be thought of as \\emph{reversible}\nif it is unitary valued.\nAs this need not be the case, the question arises whether and in what sense\none can \\emph{approximate} $T$ via unitary systems.\n\nIn\n \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\n and\n \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\na strong notion of convergence is used for the approximants.\nIn the literature a similar weak notion of convergence for $\\Cnought$-semigroups is studied\n(\\cf \\exempli\n \\cite{Krol2009,Eisner2008kato,Eisnersereny2009catThmStableSemigroups,Dahya2022weakproblem,Dahya2022complmetrproblem}\n and\n \\cite[\\S{}III.6]{Eisner2010buchStableOpAndSemigroups}\n in the case of Hilbert spaces%\n).\nThese notions are defined as follows:\nLet\n $d\\in\\naturals$,\n $\\BanachRaum$ be a Banach space,\n $\\{T_{i}\\}_{i=1}^{d}$\n be a commuting family of $\\Cnought$-semigroups on $\\BanachRaum$,\n and\n $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha\\in\\Lambda}$\n be a net of commuting families of $\\Cnought$-semigroups on $\\BanachRaum$.\nWe say that\n $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha\\in\\Lambda}$\n converges to $\\{T_{i}\\}_{i=1}^{d}$\n \\highlightTerm{\\wrt the \\topSOT-toplogy}\n (\\respectively \\highlightTerm{\\wrt the \\topWOT-toplogy})\n \\highlightTerm{uniformly on compact subsets of $\\realsNonNeg^{d}$},\nif\n for all $\\xi\\in\\BanachRaum$\n (\\respectively for all $\\xi\\in\\BanachRaum$ and $\\eta\\in\\BanachRaum^{\\prime}$)\n and all compact $L\\subseteq\\realsNonNeg^{d}$\nit holds that\n ${\n \\sup_{\\mathbf{t}\\in L}\n \\norm{\n (\n \\prod_{i=1}^{d}\n T^{(\\alpha)}_{i}(t_{i})\n -\n \\prod_{i=1}^{d}\n T_{i}(t_{i})\n )\n \\:\\xi\n }\n \\underset{\\alpha}{\\longrightarrow}\n 0\n }$\n(\\respectively\n ${\n \\sup_{\\mathbf{t}\\in L}\n \\abs{\\brkt{\n (\n \\prod_{i=1}^{d}\n T^{(\\alpha)}_{i}(t_{i})\n -\n \\prod_{i=1}^{d}\n T_{i}(t_{i})\n )\n \\:\\xi\n }{\\eta}}\n \\underset{\\alpha}{\\longrightarrow}\n 0\n }$\n).\nFor classical systems on Hilbert spaces,\nusing the concepts in \\S{}\\ref{sec:introduction:monoids:sig:article-stochastic-raj-dahya}\nwe may generalise the weak notion of convergence to classical dynamical systems\nin general in the following natural manner:\n\n\\begin{defn}[Weak topologies]\n\\makelabel{defn:weak-convergence-over-exact-uniform-ptwise:sig:article-stochastic-raj-dahya}\n Let\n $G$ be a topological group\n $M \\subseteq $ a submonoid,\n $\\HilbertRaum$ a Hilbert space,\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n an \\topSOT-continuous homomorphism\n and\n ${(T^{(\\alpha)} \\colon M \\to \\BoundedOps{\\HilbertRaum})_{\\alpha\\in\\Lambda}}$\n a net of \\topSOT-continuous homomorphisms.\n We say that\n $(T^{(\\alpha)})_{\\alpha\\in\\Lambda}$\n converges to\n $T$\n\n \\begin{kompaktenum}{(i)}\n \\item\\punktlabel{1}\n \\highlightTerm{exactly weakly},\n if for each\n $\\xi,\\eta\\in\\HilbertRaum$\n there exists an index $\\alpha_{0}\\in\\Lambda$\n such that for all\n $x \\in M$\n and\n $\\alpha \\geq \\alpha_{0}$\n\n \\vspace{-3\\parskip}\n $$\n \\brkt{T^{(\\alpha)}(x)\\xi}{\\eta}\n =\n \\brkt{T(x)\\xi}{\\eta};\n $$\n \\item\\punktlabel{2}\n \\highlightTerm{uniformly weakly},\n if for all\n $\\xi,\\eta\\in\\HilbertRaum$\n and\n compact $L \\subseteq M$\n\n \\vspace{-3\\parskip}\n $$\n \\sup_{\\mathbf{t}\\in L}\n \\abs{\n \\brkt{T^{(\\alpha)}(x)\\xi}{\\eta}\n -\n \\brkt{T(x)\\xi}{\\eta}\n }\n \\underset{\\alpha}{\\longrightarrow}\n 0;\n $$\n \\item\\punktlabel{3}\n \\highlightTerm{pointwise weakly},\n if for all\n $\\xi,\\eta\\in\\HilbertRaum$\n and\n $x \\in M$\n\n \\vspace{-3\\parskip}\n $$\n \\abs{\n \\brkt{T^{(\\alpha)}(x)\\xi}{\\eta}\n -\n \\brkt{T(x)\\xi}{\\eta}\n }\n \\underset{\\alpha}{\\longrightarrow}\n 0.\n $$\n \\end{kompaktenum}\n\n \\noindent\n If each $T^{(\\alpha)} = U^{(\\alpha)}\\restr{M}$,\n where ${U^{(\\alpha)} \\colon G \\to \\BoundedOps{\\HilbertRaum}}$\n is an \\topSOT-continuous unitary representation of $G$ on $\\HilbertRaum$,\n we say that $T$ has an\n \\highlightTerm{exact}\n (%\n \\respectively\n \\highlightTerm{uniform}\n \\respectively\n \\highlightTerm{pointwise}%\n )\n \\highlightTerm{weak unitary approximation},\n if \\punktcref{1}\n (%\n \\respectively\n \\punktcref{2}\n \\respectively\n \\punktcref{3}%\n )\n holds.\n\\end{defn}\n\n\\begin{defn}[Regular weak topologies]\n\\makelabel{defn:regular-weak-convergence-over-exact-uniform-ptwise:sig:article-stochastic-raj-dahya}\n Let\n $(G,M,\\cdot^{+})$ be a positivity structure,\n $\\HilbertRaum$ a Hilbert space,\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n an \\topSOT-continuous homomorphism\n and\n ${(T^{(\\alpha)} \\colon M \\to \\BoundedOps{\\HilbertRaum})_{\\alpha\\in\\Lambda}}$\n a net of \\topSOT-continuous homomorphisms.\n We say that\n $(T^{(\\alpha)})_{\\alpha\\in\\Lambda}$\n converges to\n $T$\n\n \\begin{kompaktenum}{(i)}\n \\item\\punktlabel{1}\n \\highlightTerm{exactly regularly weakly},\n if for each\n $\\xi,\\eta\\in\\HilbertRaum$\n there exists an index $\\alpha_{0}\\in\\Lambda$\n such that for all\n $x \\in G$\n and\n $\\alpha \\geq \\alpha_{0}$\n\n \\vspace{-3\\parskip}\n $$\n \\brkt{T^{(\\alpha)}(x^{-})^{\\ast}T^{(\\alpha)}(x^{+})\\xi}{\\eta}\n =\n \\brkt{T(x^{-})^{\\ast}T(x^{+})\\xi}{\\eta};\n $$\n \\item\\punktlabel{2}\n \\highlightTerm{uniformly regularly weakly},\n if for all\n $\\xi,\\eta\\in\\HilbertRaum$\n and\n compact $L \\subseteq G$\n\n \\vspace{-3\\parskip}\n $$\n \\sup_{x \\in L}\n \\abs{\n \\brkt{T^{(\\alpha)}(x^{-})^{\\ast}T^{(\\alpha)}(x^{+})\\xi}{\\eta}\n -\n \\brkt{T(x^{-})^{\\ast}T(x^{+})\\xi}{\\eta}\n }\n \\underset{\\alpha}{\\longrightarrow}\n 0;\n $$\n \\item\\punktlabel{3}\n \\highlightTerm{pointwise regularly weakly},\n if for all\n $\\xi,\\eta\\in\\HilbertRaum$\n and\n $x \\in G$\n\n \\vspace{-3\\parskip}\n $$\n \\abs{\n \\brkt{T^{(\\alpha)}(x^{-})^{\\ast}T^{(\\alpha)}(x^{+})\\xi}{\\eta}\n -\n \\brkt{T(x^{-})^{\\ast}T(x^{+})\\xi}{\\eta}\n }\n \\underset{\\alpha}{\\longrightarrow}\n 0.\n $$\n \\end{kompaktenum}\n\n \\noindent\n If each $T^{(\\alpha)} = U^{(\\alpha)}\\restr{M}$,\n where ${U^{(\\alpha)} \\colon G \\to \\BoundedOps{\\HilbertRaum}}$\n is an \\topSOT-continuous unitary representation of $G$ on $\\HilbertRaum$,\n we say that $T$ has an\n \\highlightTerm{exact}\n (%\n \\respectively\n \\highlightTerm{uniform}\n \\respectively\n \\highlightTerm{pointwise}%\n )\n \\highlightTerm{regular weak unitary approximation},\n if \\punktcref{1}\n (%\n \\respectively\n \\punktcref{2}\n \\respectively\n \\punktcref{3}%\n )\n holds.\n\\end{defn}\n\nClearly, exact (regular) weak approximations\nare uniform (regular) weak approximations,\nwhich in turn are pointwise (regular) weak approximations.\nIn the case of\n $(G,M)=(\\reals^{d},\\realsNonNeg^{d})$,\nif $d=1$, then\neach \\emph{regular}-notion of convergence clearly coincides\nwith the corresponding notion without the \\emph{regular} prefix.\nIn this special case, the following result is known:\n\n\\begin{thm}[Kr\\'ol, 2009]\n\\makelabel{thm:unitary-approx:one-param:krol:sig:article-stochastic-raj-dahya}\n Let $T$ be a contractive $\\Cnought$-semigroup\n on an infinite dimensional Hilbert space $\\HilbertRaum$.%\n \\footref{ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya}\n Then $T$ has a uniform weak unitary approximation.\n\\end{thm}\n\nFor a proof see \\cite[Theorem~2.1 and Remark~2.3]{Krol2009},\nin which Kr\\'ol constructs the unitary approximants directly from a regular unitary dilation of $T$.\nIn this reference, it is questioned whether a proof is possible without reliance upon dilations\n(\\cf \\cite[Remark~2.2]{Krol2009}).\nOur results partially address this by showing\nthat the existence of simultaneous (regular) dilations is necessary.\nThus any dilation-free proof of\n \\Cref{thm:unitary-approx:one-param:krol:sig:article-stochastic-raj-dahya}\nmight necessarily involve some characterisation of (regular) unitary dilations.\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{thm}[Characterisation of weak unitary approximations]\n\\makelabel{thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}\n Let\n $G$ be a locally compact topological group\n and\n $M \\subseteq G$ be an $e$-joint closed submonoid.\n Further let\n $\\HilbertRaum$ be a Hilbert space\n and\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n be an \\topSOT-continuous homomorphism.\n If $G$ contains a dense subspace $D \\subseteq G$\n with $\\card{D} \\leq \\mathop{\\textup{dim}}(\\HilbertRaum)$,%\n \\footref{ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya}\n then the following are equivalent:\n\n \\begin{kompaktenum}{\\bfseries (a)}\n \\item\\punktlabel{1}\n The classical system $T$ has a unitary dilation.\n \\item\\punktlabel{2}\n The classical system $T$ has an exact weak unitary approximation.\n \\item\\punktlabel{3}\n The classical system $T$ has a uniform weak unitary approximation.\n \\end{kompaktenum}\n\n \\noindent\n Without the above assumption on the dimension of $\\HilbertRaum$,\n \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n hold.\n\\end{thm}\n\\end{schattierteboxdunn}\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{thm}[Characterisation of regular weak unitary approximations]\n\\makelabel{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n Let\n $(G,M,\\cdot^{+})$ be a positivity structure,\n where\n $G$ is a topological group\n and\n $M \\subseteq G$ is a submonoid.%\n \\footref{ft:approx-thm:G-not-locally-compact:sig:article-stochastic-raj-dahya}\n Further let\n $\\HilbertRaum$ be a Hilbert space\n and\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n be an \\topSOT-continuous homomorphism.\n If $\\HilbertRaum$ is infinite dimensional\n and $G$ contains a dense subspace $D \\subseteq G$\n with $\\card{D} \\leq \\mathop{\\textup{dim}}(\\HilbertRaum)$,%\n \\footref{ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya}\n then the following are equivalent:\n\n \\begin{kompaktenum}{\\bfseries (a)}\n \\item\\punktlabel{1}\n The classical system $T$ has a regular unitary dilation.\n \\item\\punktlabel{2}\n The classical system $T$ has an exact regular weak unitary approximation.\n \\item\\punktlabel{3}\n The classical system $T$ has a uniform regular weak unitary approximation.\n \\item\\punktlabel{4}\n The classical system $T$ has a pointwise regular weak unitary approximation.\n \\end{kompaktenum}\n\n \\noindent\n Without the above assumption on the dimension of $\\HilbertRaum$,\n \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n hold.\n\\end{thm}\n\\end{schattierteboxdunn}\n\n\\footnotetext[ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya]{\n In the case of separable topological groups,\n \\exempli $G=\\reals^{d}$, $d\\in\\naturals$,\n this holds as soon as $\\HilbertRaum$ is infinite dimensional.\n}\n\n\\footnotetext[ft:approx-thm:G-not-locally-compact:sig:article-stochastic-raj-dahya]{\n Note that we neither require $G$ to be locally compact\n nor $M$ to be a measurable subset in this theorem!\n}\n\n\\Cref{%\n thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n}\nprovide a sharp distinction between the two notions of dilation.\nBy\n \\Cref{%\n e.g.:monoids:e-joint:sig:article-stochastic-raj-dahya,%\n e.g.:monoids:positivity:sig:article-stochastic-raj-dahya,%\n }\nthese results are immediately applicable\nto commuting families of $\\Cnought$-semigroups\nas well as non-commuting families satisfying the \\emph{canonical commutation relations} (CCR) in the Weyl form\n(see \\Cref{e.g.:non-commuting-family-heisenberg-d-C:sig:article-stochastic-raj-dahya}).\n\nAs an example, applying these characterisations to\n $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$\nfor any $d \\geq 2$ and infinite dimensional Hilbert space $\\HilbertRaum$,\nwe shall construct commuting families of $d$ contractive $\\Cnought$-semigroups on $\\HilbertRaum$\nthat admit no unitary approximations\n(see \\Cref{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya}).\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Notation]{Notation}\n\\label{sec:introduction:notation:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn this paper we fix the following notation and conventions:\n\n\\begin{kompaktitem}\n \\item\n We write\n $\\naturalsPos = \\{1,2,\\ldots\\}$,\n $\\naturalsZero = \\{0,1,2,\\ldots\\}$,\n $\\realsNonNeg = \\{r\\in\\reals \\mid r\\geq 0\\}$,\n $\\realsPos = \\{r\\in\\reals \\mid r > 0\\}$,\n and\n $\\Torus = \\{z\\in\\complex \\mid \\abs{z} = 1\\}$\n (unit circle in the complex plane).\n To distinguish from indices $i$ we use $\\iunit$ for the imaginary unit $\\sqrt{-1}$.\n \\item\n We write elements of product spaces in bold\n and denote their components in light face fonts with appropriate indices,\n \\exempli the $i$\\textsuperscript{th} components of\n $\\mathbf{t} \\in \\realsNonNeg^{n}$\n and\n $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{n}\\Lambda_{i}$\n are denoted\n $t_{i}$ and $\\alpha_{i}$\n respectively.\n \\item\n In some instances we shall work with concrete constructions of approximants of\n $\\Cnought$-semigroups or families thereof\n (\\exempli the Hille- and Yosida-approximants).\n In such cases we use\n $\\lambda \\in \\realsPos$\n and\n $\\boldsymbol{\\lambda} \\in \\realsPos^{d}$\n to index the approximants.\n In others instances we work with the generalisation: \\highlightTerm{expectation-approximants}\n (defined below in \\Cref{defn:expectation-approximants:sig:article-stochastic-raj-dahya}).\n To indicate the abstract setting,\n $\\alpha \\in \\Lambda$\n and\n $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$\n are used to index the approximants.\n \\item\n For bounded operators, $S$, over a Banach space,\n $S^{\\prime}$ denotes the adjoint (dual) operator.\n For bounded operators, $S$, over a Hilbert space,\n $S^{\\ast}$ denotes the Hermitian adjoint.\n \\item\n For a measure (or probability) space $(X,\\Sigma,\\mu)$,\n a measurable space $(Y,S)$,\n and\n a measurable function ${f \\colon X \\to Y}$,\n the push-forward measure, $\\PushForward{f}{\\mu}$,\n which we denote $\\mu_{f}$,\n is the measure (\\respectively probability measure) on $(Y, S)$\n defined by\n $\\PushForward{f}{\\mu}[B] = \\mu[f^{-1}(B)]$\n for all measurable $B \\subseteq Y$.\n \\item\n For a probability distribution, $\\Gamma$, over a set $X$\n we write ${\\theta \\distributedAs \\Gamma}$\n to denote that $\\theta$ is an $X$-valued random variable (\\randomvar)\n with distribution $\\Gamma$.\n \\item\n For $t\\in\\reals$ the distribution $\\delta_{t}$ denotes the point distribution concentrated in $t$.\n \\item\n For $\\lambda\\in\\realsPos$\n we denote with $\\DistExp{\\lambda}$\n the exponential distribution with \\highlightTerm{rate} $\\lambda$.\n For ${\\theta \\distributedAs \\DistExp{\\lambda}}$\n it holds that\n ${\n \\Prob_{\\theta}[B] = \\int_{B}\\lambda e^{\\lambda s}\\:\\dee s\n }$\n for all measurable $B \\subseteq \\realsNonNeg$.\n \\item\n For $\\lambda\\in\\realsNonNeg$ and $c\\in\\realsPos$\n we denote with $\\DistPoissScale{\\lambda}{c}$\n the distribution of a Poisson distributed \\randomvar scaled by $c$.\n For $\\theta \\distributedAs \\DistPoissScale{\\lambda}{c}$\n it holds that\n ${\n \\Prob[\\theta=cn] = \\frac{\\lambda^{n}}{n!}e^{-\\lambda}\n }$\n for $n\\in\\naturals$ with the convention that $0^{0} \\colonequals 0$.\n In particular, $\\DistPoissScale{\\lambda}{c}=\\delta_{0}$ if $\\lambda=0$.\n \\item\n For $\\lambda\\in\\realsPos$ and $t\\in\\realsNonNeg$,\n we denote with $\\DistPoissAux{t}{\\lambda}$\n an \\highlightTerm{auxiliary Poisson process}\n with \\highlightTerm{rate} $\\lambda$\n over a \\highlightTerm{time duration} $t$\n (defined below in \\S{}\\ref{sec:stochastic:distributions:sig:article-stochastic-raj-dahya},\n see also \\cite[\\S{}4]{Chung1962exp}).\n \\item\n For a (unital) {($C^{\\ast}$-)algebra} $\\mathcal{A}$,\n $M_{n}(\\mathcal{A})$ denotes the (unital) matrix {($C^{\\ast}$-)algebra}\n of $\\mathcal{A}$-valued $n \\times n$-matrices\n for $n\\in\\naturals$.\n If $\\mathcal{A}$ is an algebra of operators over some Hilbert space, $\\HilbertRaum$,\n then the elements of $M_{n}(\\mathcal{A})$\n are viewed as operators acting on $\\bigoplus_{i=1}^{n}\\HilbertRaum$,\n and we have\n $\n \\brkt{(a_{ij})_{ij}\\oplus_{i=1}^{d}\\xi_{i}}{\\oplus_{i=1}^{d}\\eta_{i}}\n = \\sum_{ij}\n \\brkt{a_{ij}\\xi_{j}}{\\eta_{i}}\n $\n for \\usesinglequotes{matrices} $(a_{ij})_{ij} \\in M_{n}(\\mathcal{A})$\n and vectors\n $\n \\oplus_{i=1}^{n}\\xi_{i},\\oplus_{i=1}^{n}\\eta_{i}\n \\in \\bigoplus_{i=1}^{n}\\HilbertRaum\n $.\n \\item\n A map ${\\Psi \\colon \\mathcal{A} \\to \\mathcal{B}}$\n between (subalgebras of) $C^{\\ast}$-algebras is called\n \\emph{completely bounded} if\n $\n \\normCb{\\Psi}\n \\colonequals\n \\sup_{n\\in\\naturalsPos}\\norm{\\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}}}\n < \\infty\n $,\n and \\emph{completely positive} if\n $\n \\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}}\n $\n is positive for all $n\\in\\naturals$.\n Here\n ${\\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}} \\colon M_{n}(\\mathcal{A}) \\to M_{n}(\\BoundedOps{\\HilbertRaum})}$\n is defined by\n $(\\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}})((a_{ij})_{ij}) = (\\Psi(a_{ij}))_{ij} \\in M_{n}(\\BoundedOps{\\HilbertRaum})$\n for each\n $(a_{ij})_{ij} \\in M_{n}(\\mathcal{A})$\n and\n $n\\in\\naturals$\n (%\n see\n \\cite[Chapter~1]{Paulsen1986bookCBmapsAndDilations},\n \\cite[Chapter~3]{Pisier2001bookCBmaps}%\n ).\n\\end{kompaktitem}\n\nFor a Banach space $\\BanachRaum$,\na measure space $(X,\\Sigma,\\mu)$\nand an operator-valued function ${T \\colon X \\to \\BoundedOps{\\BanachRaum}}$,\nfor which ${T(\\cdot)\\xi \\colon X \\to \\BanachRaum}$ is\n\\highlightTerm{strongly measurable} for each $\\xi\\in\\BanachRaum$,\nthe integral ${\\sotInt_{X} T\\:\\dee\\mu}$, when it exists,\ndenotes the unique bounded operator $\\tilde{T}\\in\\BoundedOps{\\BanachRaum}$\nthat satisfies\n ${\\tilde{T}\\xi = \\int_{X} T(\\cdot)\\xi\\:\\dee\\mu}$\nfor all $\\xi\\in\\BanachRaum$,\nwhere $\\int_{X} T(\\cdot)\\xi\\:\\dee\\mu$\nis computed as a Bochner-integral.\nThis holds, for example, if $X$ is a locally compact Polish space\n(\\exempli $\\realsNonNeg^{d}$ for some $d\\in\\naturals$),\nand $T$ is contractive and \\topSOT-continuous\n(\\exempli a product of contractive $\\Cnought$-semigroups).\nIf $(\\Omega,\\Sigma,\\Prob)$ is a probability space\nand ${\\tau \\colon \\Omega \\to X}$ is an $X$-valued \\randomvar (\\idest a measurable function),\nwe refer to\n $\n \\Expected[T(\\theta)]\n = \\sotInt_{\\omega\\in\\Omega} T(\\theta(\\omega))\\:\\Prob(\\dee\\omega)\n = \\sotInt_{t \\in X} T(t)\\:\\Prob_{\\theta}(\\dee t)\n $,\nwhen it exists,\nas the \\highlightTerm{expectation} (computed strongly via Bochner-integrals).\nIf $X$ is a locally compact Polish space\nand $T$ is a contractive \\topSOT-continuous function\nthen the expectation exists.\n\nThe existence and properties of Bochner-integrals\n(including linearity, convexity and triangle inequalities, Fubini's theorem for products, \\etcetera)\nas well as the validity of various computations with Bochner-integrals and expectations\nused in the rest of this paper\ncan be found in or readily derived from the literature.\nWe refer the reader in particular to\n \\cite[\\S{}3.7, Theorems~3.7.4--6, and Theorems~3.7.12--13]{Hillephillips1957faAndSg},\n \\cite[\\S{}II.2, Theorem~2, and Theorem~4]{DiestelUhl1977VectorMeasures},\n \\cite[\\S{}C.1--4]{EngelNagel2000semigroupTextBook},\n and\n \\cite[\\S{}1.1.c--\\S{}1.2.a]{HytNeervanMarkLutz2016bookVol1}.\nFor example, using Fubini's theorem one can derive that\n $\n \\prod_{i=1}^{n}\\Expected[T_{i}(\\theta_{i})]\n = \\Expected[\\prod_{i=1}^{n}T_{i}(\\theta_{i})]\n $\nfor $n\\in\\naturals$,\nindependent $\\realsNonNeg$-valued \\randomvar's\n $\\theta_{1},\\theta_{2},\\ldots,\\theta_{n}$\nand $\\Cnought$-semigroups $T_{1},T_{2},\\ldots,T_{n}$ on a Banach space $\\BanachRaum$\nwhich are uniformly bounded on the essential ranges of\n$\\theta_{1},\\theta_{2},\\ldots,\\theta_{n}$ respectively\n(\\exempli contractive semigroups).\nWe shall take advantage of this computation throughout.\nFurther fundamental applications of Bochner-integrals\nin the context of $\\Cnought$-semigroups\ncan be found \\exempli in \\cite{Chung1962exp,Dunfordschwartz1988BookLinOpI,Reissig2005abstractresolvent}.\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Stochastic presentation of classical approximants]{Stochastic presentation of classical approximants}\n\\label{sec:stochastic:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn this section we provide standalone results for $\\Cnought$-semigroups\nover Banach spaces and then for (commuting) families.\nWe assume basic knowledge of stochastic processes\nas well as Poisson and exponential distributions.\nWe shall also rely on the theory of Bochner-integrals,\nin particular those that occur in the integral representations of\npowers of resolvents of unbounded generators of $\\Cnought$-semigroups.\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Semigroups of distributions]{Semigroups of distributions}\n\\label{sec:stochastic:distributions:sig:article-stochastic-raj-dahya}\n\n\\noindent\nA family $(\\Gamma(t))_{t\\in\\realsNonNeg}$ of parameterised distributions\nof $\\reals$-valued random variables (\\randomvar's)\nshall be called a \\highlightTerm{continuous semigroup of distributions} if\n\n\\begin{kompaktenum}{(i)}\n \\item\n $\\Gamma(0)$ is the point distribution concentrated in $0$ (\\idest $\\delta_{0}$);\n \\item\n $\\theta_{1} + \\theta_{2} \\distributedAs \\Gamma(s + t)$\n for $s,t\\in\\realsNonNeg$\n and any independent \\randomvar's\n $\\theta_{1} \\distributedAs \\Gamma(s)$\n and\n $\\theta_{2} \\distributedAs \\Gamma(t)$;\n and\n \\item\n letting $\\theta_{t} \\distributedAs \\Gamma(t)$\n and $\\mu_{t}$ be the associated probability distribution\n for $t\\in\\realsNonNeg$,\n we have\n ${\\mu_{t} \\longrightarrow \\mu_{0} = \\delta_{0}}$\n weakly for ${\\realsPos \\ni t \\longrightarrow 0}$,\n \\idest\n ${\\Expected[f(\\theta_{t})] \\longrightarrow \\Expected[f(\\theta_{0})] = f(0)}$\n for all bounded continuous functions $f\\in\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}{\\realsNonNeg}$\n\\end{kompaktenum}\n\n\\noindent\n(\\cf\n \\cite[Definition~14.46 and Example~17.7]{Klenke2008probTheory}\n and\n \\cite[\\S{}17.E]{Kechris1995BookDST}%\n).\nSimple examples of this include\n $(\\delta_{t})_{t\\in\\realsNonNeg}$,\n $(\\DistPoissScale{\\lambda t}{c})_{t\\in\\realsNonNeg}$\n for $c\\in\\reals$, $\\lambda\\in\\realsPos$,\n and\n $(\\mathcal{N}(\\mu t,\\sigma^{2}t))_{t\\in\\realsNonNeg}$\n (the normal distributions)\n for $\\mu\\in\\reals$, $\\sigma\\in\\realsNonNeg$\n with the convention that\n $\\mathcal{N}(0,0)$ denotes the point distribution $\\delta_{0}$\n(\\cf \\cite[Corollary~15.13]{Klenke2008probTheory}).\n\nWe now construct a further family of parameterised distributions.\nFor $\\lambda \\in (0,\\:\\infty)$ we construct a random distribution\nvia two independent homogeneous point Poisson processes (PPP) as follows\n(depicted in \\Cref{fig:example-auxiliary-poisson:sig:article-stochastic-raj-dahya}):\n\n\\begin{kompaktenum}{\\bfseries 1.}\n \\item\n Let\n $\n \\tilde{\\tau}_{0}, \\tilde{\\tau}_{1}, \\tilde{\\tau}_{2},\\ldots,\n \\tau_{0}, \\tau_{1}, \\tau_{2},\\ldots\n \\distributedAs\n \\DistExp{\\lambda}\n $\n be independent identically distributed (\\iid) \\randomvar's.\n \\item\n For each $n\\in\\naturalsZero$ set\n $\n \\tau_{, draw=black, line width=0.5pt, fill=none]\n ({-0.1 * \\hunit}, {0 * \\vunit}) -- ({2.6 * \\hunit}, {0 * \\vunit})\n node [below, align=center]{\\footnotesize time};\n \\end{tikzpicture}\n \\caption{%\n 1\\textsuperscript{st} homogeneous PPP with rate $\\lambda$,\n where $N_{t}$ counts the number of \\usesinglequotes{events} in $[0,\\:t)$.\n }\n \\label{fig:example-auxiliary-poisson:a:sig:article-stochastic-raj-dahya}\n \\end{subfigure}\n\n \\begin{subfigure}[m]{\\textwidth}\n \\centering\n \\begin{tikzpicture}[node distance=1cm, thick]\n \\pgfmathsetmacro\\hunit{3}\n \\pgfmathsetmacro\\vunit{1}\n\n \n \\draw [draw=none, fill=none]\n ({-0.1 * \\hunit}, {0.5 * \\vunit})\n -- ({2.6 * \\hunit}, {0.5 * \\vunit})\n -- ({2.6 * \\hunit}, {-0.5 * \\vunit})\n -- ({-0.1 * \\hunit}, {-0.5 * \\vunit})\n -- cycle;\n\n \n \\draw [draw=none, fill=drawing_light_grey]\n ({0 * \\hunit}, {0 * \\vunit})\n -- ({2.5 * \\hunit}, {0 * \\vunit})\n -- ({2.5 * \\hunit}, {0.5 * \\vunit})\n -- ({0 * \\hunit}, {0.5 * \\vunit})\n -- cycle;\n\n \n \\draw [draw=blue, line width=0.5pt, fill=none] ({0.1704947206981121 * \\hunit}, {0 * \\vunit}) -- ({0.1704947206981121 * \\hunit}, {0.5 * \\vunit});\n \\draw [draw=blue, line width=0.5pt, fill=none] ({0.986362751410455 * \\hunit}, {0 * \\vunit}) -- ({0.986362751410455 * \\hunit}, {0.5 * \\vunit});\n \\draw [draw=blue, line width=0.5pt, fill=none] ({1.1292144647056632 * \\hunit}, {0 * \\vunit}) -- ({1.1292144647056632 * \\hunit}, {0.5 * \\vunit});\n \\draw [draw=blue, line width=0.5pt, fill=none] ({1.2332491071729783 * \\hunit}, {0 * \\vunit}) -- ({1.2332491071729783 * \\hunit}, {0.5 * \\vunit});\n \\draw [draw=blue, line width=0.5pt, fill=none] ({1.4080395043214222 * \\hunit}, {0 * \\vunit}) -- ({1.4080395043214222 * \\hunit}, {0.5 * \\vunit});\n \\draw [draw=blue, line width=0.5pt, fill=none] ({1.5578881324962115 * \\hunit}, {0 * \\vunit}) -- ({1.5578881324962115 * \\hunit}, {0.5 * \\vunit});\n \\draw [draw=black, line width=0.5pt, fill=none] ({2.0546540974785676 * \\hunit}, {0 * \\vunit}) -- ({2.0546540974785676 * \\hunit}, {0.5 * \\vunit});\n\n \n \\draw [decorate, decoration={brace, amplitude=5pt}]\n ({0 * \\hunit}, {0.55 * \\vunit}) -- ({1.5578881324962115 * \\hunit}, {0.55 * \\vunit})\n node[midway, above=5pt, align=center]{\\footnotesize $N_{t}$};\n\n \n \\draw [draw=black, line width=2pt, fill=none]\n ({0 * \\hunit}, {-0.05 * \\vunit})\n node [label=below:{\\footnotesize $0$}]{}\n -- ({0 * \\hunit}, {0.1 * \\vunit});\n\n \n \\draw [draw=blue, line width=1pt, fill=none]\n ({1.5578881324962115 * \\hunit}, {-0.05 * \\vunit})\n node [label=below:{\\footnotesize $\\theta$}]{}\n -- ({1.5578881324962115 * \\hunit}, {0.5 * \\vunit});\n\n \n \\draw [->, draw=black, line width=0.5pt, fill=none]\n ({-0.1 * \\hunit}, {0 * \\vunit}) -- ({2.6 * \\hunit}, {0 * \\vunit})\n node [below, align=center]{\\footnotesize time};\n \\end{tikzpicture}\n \\caption{%\n 2\\textsuperscript{nd} homogeneous PPP with rate $\\lambda$,\n independent of 1\\textsuperscript{st} PPP,\n but with $N_{t}$ determined by (a).\n }\n \\label{fig:example-auxiliary-poisson:b:sig:article-stochastic-raj-dahya}\n \\end{subfigure}\n\n \\caption{%\n Visualisation of the construction of an \\highlightTerm{auxiliary Poisson process},\n $\\theta \\distributedAs \\DistPoissAux{t}{\\lambda}$.\n }\n \\label{fig:example-auxiliary-poisson:sig:article-stochastic-raj-dahya}\n\\end{figure}\n\n\nFor each $\\lambda\\in\\realsPos$ and $t\\in\\realsNonNeg$\nlet $\\DistPoissAux{t}{\\lambda}$ denote the distribution of $\\theta_{t}$\nconstructed as above.\nWe refer to any \\randomvar distributed as $\\DistPoissAux{t}{\\lambda}$\nas an \\highlightTerm{auxiliary Poisson process}\nwith \\highlightTerm{rate} $\\lambda$\nover a \\highlightTerm{time duration} $t$.\nWe observe some basic properties of auxiliary Poisson processes.\n\n\\begin{prop}\n\\makelabel{prop:characteristic-function:aux-poisson-process:sig:article-stochastic-raj-dahya}\n Let $\\lambda\\in\\realsPos$ and $t\\in\\realsNonNeg$.\n The characteristic function\n of a $\\DistPoissAux{t}{\\lambda}$-distributed \\randomvar%\n \\footnote{\n For an $\\reals$-valued \\randomvar $X$\n the characteristic function is defined as the map\n ${\\altvarphi \\colon \\reals \\ni \\omega \\mapsto \\Expected[e^{\\iunit \\omega X}}]$\n (\\cf \\cite[Definition~15.7]{Klenke2008probTheory}).\n }\n is given by\n $\n \\altvarphi_{t, \\lambda}(\\omega)\n = e^{\\frac{\\iunit\\omega}{\\lambda - \\iunit\\omega} \\lambda t}\n $\n for all $\\omega \\in \\reals$.\n Moreover, the mean and variance of\n $\\DistPoissAux{t}{\\lambda}$-distributed \\randomvar's\n are $t$ and $\\frac{2t}{\\lambda}$ respectively.\n\\end{prop}\n\n \\begin{proof}\n Let $\\theta \\distributedAs \\DistPoissAux{t}{\\lambda}$.\n Without loss of generality, we may assume that $\\theta$\n is given by the above construction, \\idest\n $\\theta = \\theta_{t} = \\tau_{ 0$ be arbitrary.\n Since $T$ is \\topSOT-continuous,\n ${\n \\sup_{s\\in[0,\\:\\delta]}\n \\norm{(T(s) - \\onematrix)\\xi}\n < \\varepsilon\n }$\n for some $\\delta > 0$.\n Thus\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\norm{(T^{(\\alpha)}(t) - T^{(\\alpha)}(0))\\xi}\n &= &\\normLong{\\Big(\\Expected[T(\\theta^{(\\alpha)}_{t})] - \\onematrix\\Big)\\xi}\\\\\n &= &\\normLong{\n \\displaystyle\n \\int_{s\\in\\realsNonNeg}\n (T(s) - \\onematrix)\\xi\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\n }\\\\\n &\\leq &\\displaystyle\n \\int_{s\\in\\realsNonNeg}\n \\norm{(T(s) - \\onematrix)\\xi}\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n &= &\\displaystyle\n \\int_{s\\in[0,\\:\\delta]}\n \\underbrace{\n \\normLong{(T(s) - \\onematrix)\\xi}\n }_{<\\varepsilon}\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\n + \\displaystyle\n \\int_{s\\in(\\delta,\\:\\infty)}\n \\underbrace{\n \\normLong{(T(s) - \\onematrix)\\xi}\n }_{\\leq 2\\norm{\\xi}}\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n &\\leq &\\varepsilon\\,\\Prob_{\\theta^{(\\alpha)}_{t}}[[0,\\:\\delta]]\n + 2\\norm{\\xi}\\,\\Prob_{\\theta^{(\\alpha)}_{t}}[(\\delta,\\:\\infty)]\\\\\n \\end{eqnarray*}\n\n \\noindent\n for all $t\\in\\realsNonNeg$.\n By the Portmanteau theorem (see \\cite[Theorem~17.20~v)]{Kechris1995BookDST}) and since by definition\n ${\\theta^{(\\alpha)}_{t} \\underset{t}{\\longrightarrow} \\delta_{0}}$\n and\n $\n \\delta_{0}(\\quer{(\\delta,\\:\\infty)} \\mathbin{\\setminus} \\topInterior{(\\delta,\\:\\infty)}\n = \\delta_{0}(\\{\\delta\\})\n = 0\n $,\n one has\n ${\n \\Prob_{\\theta^{(\\alpha)}_{t}}[(\\delta,\\:\\infty)]\n \\longrightarrow\n \\Prob_{\\delta_{0}}[(\\delta,\\:\\infty)]\n = 0\n }$\n for ${\\realsPos \\ni t \\longrightarrow 0}$.\n From the above inequality and since $\\varepsilon > 0$ was arbitrarily chosen,\n it follows that\n ${T^{(\\alpha)}(t)\\xi \\longrightarrow T^{(\\alpha)}(0)\\xi}$\n for ${\\realsPos \\ni t \\longrightarrow 0}$.\n So $T^{(\\alpha)}$ constitutes a contractive $\\Cnought$-semigroup.\n\n \\paragraph{Approximation:}\n Fix an arbitrary\n $\\xi\\in\\BanachRaum$\n and\n compact subset $L \\subseteq \\realsNonNeg$.\n Let $\\varepsilon > 0$ be arbitrary.\n Without loss of generality, we can assume that $L=[0,\\:a]$\n for some $a\\in\\realsPos$.\n By the \\topSOT-continuity of $T$ and compactness of $[0,\\:a+1]$,\n there exists $\\delta \\in (0,\\:1)$ such that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n \\sup_{s \\in U}\n \\norm{(T(s) - T(t))\\xi}\n < \\varepsilon\n \\end{eqnarray}\n\n \\noindent\n for all $t \\in L$,\n where ${U \\colonequals \\realsNonNeg \\cap (t-\\delta,\\:t+\\delta)}$.\n Since by definition of expectation-approximants,\n ${\\mu^{(\\alpha)}_{t} \\underset{\\alpha}{\\longrightarrow} t}$\n uniformly for $t \\in L$,\n by \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} it follows that\n ${\\sup_{t \\in L}\\norm{(T(\\mu^{(\\alpha)}_{t}) - T(t))\\xi} \\leq \\varepsilon}$\n for sufficiently large indices $\\alpha \\in \\Lambda$.\n Furthermore, by the Chebyshev-inequality (see \\exempli \\cite[Theorem~5.11]{Klenke2008probTheory}), one has that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:chebyshev:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n \\Prob_{\\theta^{(\\alpha)}_{t}}[\\realsNonNeg \\mathbin{\\setminus} U]\n \\leq \\delta^{-2}(\\sigma^{(\\alpha)}_{t})^{2}\n \\end{eqnarray}\n\n \\noindent\n for each $t \\in L$.\n Since by definition of expectation-approximants,\n ${(\\sigma^{(\\alpha)}_{t})^{2} \\underset{\\alpha}{\\longrightarrow} 0}$\n uniformly for $t \\in L$,\n by \\eqcref{eq:chebyshev:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} one has\n ${\\sup_{t \\in L}\\Prob_{\\theta^{(\\alpha)}_{t}}[\\realsNonNeg \\mathbin{\\setminus} U] \\leq \\varepsilon}$\n for sufficiently large indices $\\alpha \\in \\Lambda$.\n For sufficiently large indices\n $\\alpha \\in \\Lambda$ and all $t \\in L$\n it follows that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\norm{(T^{(\\alpha)}(t) - T(t))\\xi}\n &\\leq\n &\\norm{(T(\\mu^{(\\alpha)}_{t}) - T(t))\\xi}\n + \\norm{(T^{(\\alpha)}(t) - T(\\mu^{(\\alpha)}_{t}))\\xi}\\\\\n &=\n &\\underbrace{\n \\norm{(T(\\mu^{(\\alpha)}_{t}) - T(t))\\xi}\n }_{\\leq\\varepsilon}\n + \\normLong{\n \\Big(\\Expected[T(\\theta^{(\\alpha)}_{t})] - T(\\mu^{(\\alpha)}_{t})\\Big)\\xi\n }\\\\\n &\\leq &\\varepsilon + \\normLong{\n \\displaystyle\n \\int_{s\\in\\realsNonNeg}\n (T(s) - T(\\mu^{(\\alpha)}_{t}))\\xi\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\n }\\\\\n &\\leq &\\varepsilon\n + \\displaystyle\n \\int_{s\\in\\realsNonNeg}\n \\norm{(T(s) - T(\\mu^{(\\alpha)}_{t}))\\xi}\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n &\\leq &\\begin{array}[t]{0l}\n \\varepsilon + \\displaystyle\n \\int_{s \\in U}\n \\underbrace{\n \\normLong{(T(s) - T(t))\\xi}\n + \\normLong{(T(t) - T(\\mu^{(\\alpha)}_{t}))\\xi}\n }_{\\leq 2\\varepsilon}\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n + \\displaystyle\n \\int_{s \\in \\realsNonNeg \\mathbin{\\setminus} U}\n \\underbrace{\n \\normLong{(T(s) - T(\\mu^{(\\alpha)}_{t}))\\xi}\n }_{\\leq 2\\norm{\\xi}}\n \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n \\end{array}\\\\\n &\\leq &\\varepsilon + 2\\varepsilon\\,\\Prob_{\\theta^{(\\alpha)}_{t}}[U]\n + 2\\norm{\\xi}\\,\\underbrace{\n \\Prob_{\\theta^{(\\alpha)}_{t}}[\\realsNonNeg \\mathbin{\\setminus} U]\n }_{\\leq\\varepsilon}\n \\leq 3\\varepsilon + 2\\norm{\\xi}\\varepsilon.\\\\\n \\end{eqnarray*}\n\n \\noindent\n Since $\\xi$ and $\\varepsilon$ were arbitrarily chosen,\n it follows that\n ${T^{(\\alpha)}(t) \\longrightarrow T(t)}$\n \\wrt the \\topSOT-topology\n uniformly in $t$\n on compact subsets of $\\realsNonNeg$.\n \\end{proof}\n\n\\begin{prop}\n\\makelabel{prop:expectation-approximants:basic:commuting-contractive-families:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\BanachRaum$ be a Banach space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$ be a commuting family\n of contractive $\\Cnought$-semigroups on $\\BanachRaum$.\n Furthermore, let\n $(T^{(\\alpha)}_{i})_{\\alpha\\in\\Lambda_{i}}$\n be expectation-approximants for $T_{i}$\n for each $i\\in\\{1,2,\\ldots,d\\}$.\n Then for each\n $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d} \\Lambda_{i}$\n the family of approximants\n $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n is a commuting family of contractive $\\Cnought$-semigroups.\n Moreover,\n $(\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d})_{\\boldsymbol{\\alpha}\\in\\prod_{i=1}^{d} \\Lambda_{i}}$\n converges to $\\{T_{i}\\}_{i=1}^{d}$\n \\wrt the \\topSOT-topology\n uniformly on compact subsets of $\\realsNonNeg^{d}$.\n\\end{prop}\n\n \\begin{proof}\n \\paragraph{Commuting family:}\n By \\Cref{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya},\n the approximants $T^{(\\alpha_{i})}_{i}$\n are contractive $\\Cnought$-semigroups\n for each $i\\in\\{1,2,\\ldots,d\\}$.\n Let $\\mathbf{t}\\in\\realsNonNeg^{d}$ be arbitrary.\n Per definition of expectation-approximants\n there exist $\\realsNonNeg$-valued \\randomvar's\n $\\theta_{1},\\theta_{1},\\ldots,\\theta_{d}$\n satisfying\n ${T^{(\\alpha_{i})}_{i}(t_{i}) = \\Expected[T_{i}(\\theta_{i})]}$\n for each $i\\in\\{1,2,\\ldots,d\\}$.\n Without loss of generality, we may assume that the $\\theta_{i}$ are independent \\randomvar's.\n By independence and commutativity\n $\n \\Expected[T_{i}(\\theta_{i})]\n \\Expected[T_{j}(\\theta_{j})]\n = \\Expected[T_{i}(\\theta_{i})T_{j}(\\theta_{j})]\n = \\Expected[T_{j}(\\theta_{j})T_{i}(\\theta_{i})]\n = \\Expected[T_{j}(\\theta_{j})]\n \\Expected[T_{i}(\\theta_{i})]\n $\n for all $i,j\\in\\{1,2,\\ldots,d\\}$ with $i \\neq j$.\n Thus $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n is a commuting family of contractive $\\Cnought$-semigroups.\n\n \\paragraph{Approximation:}\n Let $L \\subseteq \\realsNonNeg^{d}$ be an arbitrary compact subset.\n Without loss of generality, one may assume $L = \\prod_{i=1}^{d}L_{i}$\n for some compact subsets $L_{i} \\subseteq \\realsNonNeg$, $i\\in\\{1,2,\\ldots,d\\}$.\n We prove by induction over $k\\in\\{1,2,\\ldots,d\\}$ that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n \\sup_{\\mathbf{t} \\in \\prod_{i=1}^{k}L_{i}}\n \\normLong{\n \\Big(\n \\prod_{i=1}^{k}\n T^{(\\alpha_{i})}_{i}(t_{i})\n -\n \\prod_{i=1}^{k}\n T_{i}(t_{i})\n \\Big)\n \\xi\n }\n \\underset{\\boldsymbol{\\alpha}}{\\longrightarrow} 0\n \\end{eqnarray}\n\n \\noindent\n for all $\\xi\\in\\BanachRaum$.\n For $k=1$, this holds by \\Cref{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya}.\n Let $1 < k \\leq d$ and assume that \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} holds for $k-1$.\n Let $\\xi\\in\\BanachRaum$ and $\\varepsilon > 0$ be arbitrary.\n Since $T_{k}$ is \\topSOT-continuous and $L_{k}$ is compact,\n there is a finite subset $F \\subseteq L_{k}$ such that\n $\\min_{t^{\\prime} \\in F}\\norm{(T_{k}(t) - T_{k}(t^{\\prime}))\\xi} < \\varepsilon$\n for each $t \\in L_{k}$.\n Let $\\mathbf{t} \\in L$ be arbitrary\n and let $t^{\\prime} \\in F$ be such\n that $\\norm{(T_{k}(t_{k}) - T_{k}(t^{\\prime}))\\xi} < \\varepsilon$.\n Since the approximants are all contractive,\n one obtains\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\normLong{\n \\Big(\n \\displaystyle\n \\prod_{i=1}^{k}\n T^{(\\alpha_{i})}_{i}(t_{i})\n -\n \\displaystyle\n \\prod_{i=1}^{k}\n T_{i}(t_{i})\n \\Big)\n \\xi\n }\n &\\leq\n &\\normLong{\n \\Big(\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T^{(\\alpha_{i})}_{i}(t_{i})\n -\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T_{i}(t_{i})\n \\Big)\n T_{k}(t^{\\prime})\n \\xi\n }\\\\\n &&+ \\underbrace{\n \\normLong{\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T^{(\\alpha_{i})}_{i}(t_{i})\n -\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T_{i}(t_{i})\n }\n }_{\\leq 2}\n \\underbrace{\n \\normLong{\n \\Big(\n T_{k}(t^{\\prime}) - T_{k}(t_{k})\n \\Big)\n \\xi\n }\n }_{< \\varepsilon}\\\\\n &&+ \\underbrace{\n \\normLong{\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T^{(\\alpha_{i})}_{i}(t_{i})\n }\n }_{\\leq 1}\n \\normLong{\n \\Big(\n T^{(\\alpha_{k})}_{k}(t_{k}) - T_{k}(t_{k})\n \\Big)\n \\xi\n }\\\\\n &\\leq\n &\\displaystyle\n \\max_{t^{\\prime\\prime} \\in F}\n \\normLong{\n \\Big(\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T^{(\\alpha_{i})}_{i}(t_{i})\n -\n \\displaystyle\n \\prod_{i=1}^{k-1}\n T_{i}(t_{i})\n \\Big)\n T_{k}(t^{\\prime\\prime})\n \\xi\n }\\\\\n &&+ 2\\varepsilon + \\norm{\n (T^{(\\alpha_{k})}_{k}(t_{k}) - T_{k}(t_{k}))\n \\xi\n }\\\\\n \\end{eqnarray*}\n\n \\noindent\n for each $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{k}\\Lambda_{i}$.\n By\n induction\n (%\n \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} applied to $k-1$\n and the finite set of vectors\n $\\{T_{k}(t^{\\prime\\prime})\\xi \\mid t^{\\prime\\prime} \\in F\\}$%\n )\n and\n \\Cref{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya}\n (applied to $T_{k}$),\n by taking $\\limsup$ over $\\boldsymbol{\\alpha}$,\n the right-hand expression is bounded by $0 + 2\\varepsilon + 0$.\n Since $\\varepsilon > 0$ was arbitrarily chosen, it follows that \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} holds.\n Hence the claim holds by induction.\n \\end{proof}\n\nWe also obtain the following auxiliary results for simple modifications of $\\Cnought$-semigroups:\n\n\\begin{prop}\n\\makelabel{prop:basic-operations:expectation-approximations:sig:article-stochastic-raj-dahya}\n Let\n $\\BanachRaum$ be a Banach space\n and\n $T$ be a contractive $\\Cnought$-semigroup on $\\BanachRaum$.\n Furthermore, let\n $(T^{(\\alpha)})_{\\alpha \\in \\Lambda}$\n be a net of expectation-approximants for $T$\n with associated distribution semigroups\n $(\\Gamma^{(\\alpha)})_{\\alpha \\in \\Lambda}$.\n Then\n for any (equivalently: every) \\randomvar\n ${\\theta\\distributedAs\\Gamma^{(\\alpha)}(t)}$,\n $t\\in\\realsNonNeg$,\n and\n $\\alpha \\in \\Lambda$\n\n \\begin{kompaktenum}{\\bfseries (i)}\n \\item\\punktlabel{1}\n $(T^{(\\alpha)}(t))^{\\prime} = \\Expected[T(\\theta)^{\\prime}]$,\n provided $\\BanachRaum$ is reflexive; and\n \\item\\punktlabel{2}\n $(T^{(\\alpha)}(t))^{\\ast} = \\Expected[T(\\theta)^{\\ast}]$,\n if $\\BanachRaum$ is a Hilbert space.\n \\end{kompaktenum}\n\n \\noindent\n Furthermore, in the case of Hille-(\\respectively Yosida-)approximants\n $(T^{(\\lambda)})_{\\lambda \\in \\realsPos}$\n it holds that\n\n \\begin{kompaktenum}{\\bfseries (i)}\n \\setcounternach{enumi}{3}\n \\item\\punktlabel{3}\n $T^{(\\lambda)}(rt) = \\Expected[T(r\\theta)]$;\n \\end{kompaktenum}\n\n \\noindent\n for $r\\in\\realsPos$\n and any (equivalently: every) \\randomvar\n ${\\theta \\distributedAs \\DistPoissScale{r\\lambda t}{\\tfrac{1}{r\\lambda}}}$\n (\\respectively ${\\theta \\distributedAs \\DistPoissAux{t}{r\\lambda}}$).\n\\end{prop}\n\n \\begin{proof}\n \\paragraph{\\punktcref{1}:}\n First observe that\n $(T^{(\\alpha)}(t)^{\\prime})_{t\\in\\realsNonNeg}$\n and\n $(T(t)^{\\prime})_{t\\in\\realsNonNeg}$\n are $\\Cnought$-semigroups\n (see \\exempli \\cite[Theorem~I.4.9]{Goldstein1985semigroups}).\n Let $\\xi\\in\\BanachRaum^{\\prime\\prime} \\cong \\BanachRaum$\n and $\\eta\\in\\BanachRaum^{\\prime}$ be arbitrary.\n It holds that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\\xi}{T^{(\\alpha)}(t)^{\\prime}\\eta}\n &= &\\brkt{T^{(\\alpha)}(t)\\xi}{\\eta}\\\\\n &= &\\brkt{\\Expected[T(\\theta)]\\xi}{\\eta}\\\\\n &= &\\brktLong{\n \\Big(\n \\displaystyle\n \\sotInt_{s\\in\\realsNonNeg}\n T(s)\n \\:\\Prob_{\\theta}(\\dee s)\n \\Big)\n \\xi\n }{\\eta}\\\\\n &= &\\displaystyle\n \\int_{s\\in\\realsNonNeg}\n \\brkt{T(s)\\xi}{\\eta}\n \\:\\Prob_{\\theta}(\\dee s)\\\\\n &= &\\displaystyle\n \\int_{s\\in\\realsNonNeg}\n \\brkt{\\xi}{T(s)^{\\prime}\\eta}\n \\:\\Prob_{\\theta}(\\dee s)\\\\\n &= &\\brktLong{\\xi}{\n \\Big(\n \\displaystyle\n \\sotInt_{s\\in\\realsNonNeg}\n T(s)^{\\prime}\n \\:\\Prob_{\\theta}(\\dee s)\n \\Big)\n \\eta\n }\\\\\n &= &\\brkt{\\xi}{\\Expected[T(\\theta)^{\\prime}]\\eta}.\\\\\n \\end{eqnarray*}\n\n \\noindent\n It follows that $T^{(\\alpha)}(t)^{\\prime} = \\Expected[T(\\theta)^{\\prime}]$.\n The expression in \\punktcref{2} can be proved analogously.\n\n \\paragraph{\\punktcref{3}:}\n First observe that\n ${T_{r} \\colonequals (T(rt^{\\prime}))_{t^{\\prime}\\in\\realsNonNeg}}$\n is a $\\Cnought$-semigroup with generator $A_{r} = r A$.\n Let $\\lambda\\in\\realsPos$.\n In the case of Hille-approximants, one has\n\n \\vspace{-3\\parskip}\n $$\n A_{r}^{(r\\lambda)}\n = (r\\lambda) \\cdot (\n T_{r}(\\tfrac{1}{r\\lambda}) - \\onematrix\n )\n = (r\\lambda) \\cdot (\n T(\\tfrac{1}{\\lambda}) - \\onematrix\n )\n = r A^{(\\lambda)}\n $$\n\n \\noindent\n and in the case of Yosida-approximants\n\n \\vspace{-3\\parskip}\n $$\n A_{r}^{(r\\lambda)}\n = (r\\lambda) \\cdot (\n (r\\lambda) \\opResolvent{A_{r}}{r\\lambda}\n - \\onematrix\n )\n = r\\lambda \\cdot (\n (r\\lambda) \\opResolvent{rA}{r\\lambda}\n - \\onematrix\n )\n = rA^{(\\lambda)}.\n $$\n\n \\noindent\n Thus in both cases\n $\n T^{(r\\lambda)}_{r}(t)\n = e^{t A_{r}^{(r\\lambda)}}\n = e^{t r A^{(\\lambda)}}\n = T^{(\\lambda)}(rt)\n $\n holds.\n Applying the properties of the\n $(r\\lambda)$\\textsuperscript{th}-Hille-approximant\n (\\respectively $(r\\lambda)$\\textsuperscript{th}-Yosida-approximant)\n of $T_{r}$ thus yields\n\n \\vspace{-3\\parskip}\n $$\n T^{(\\lambda)}(rt)\n = T^{(r\\lambda)}_{r}(t)\n = \\Expected[T_{r}(\\theta)]\n = \\Expected[T(r\\theta)]\n $$\n\n \\noindent\n for any (equivalently: every) \\randomvar\n ${\\theta \\distributedAs \\DistPoissScale{r\\lambda t}{\\tfrac{1}{r\\lambda}}}$\n (\\respectively ${\\theta \\distributedAs \\DistPoissAux{t}{r\\lambda}}$).\n \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Families of approximants as expectations]{Families of approximants as expectations}\n\\label{sec:stochastic:multi-parameter:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn oder to prove\n \\eqcref{it:3:thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\eqcref{it:4:thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n of\n \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\nwe shall rely on results about the expectations of products.\n\n\\begin{prop}\n\\makelabel{prop:expectation-approximants:product:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\BanachRaum$ be a Banach space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$ be a (not necessarily commuting) family of\n contractive $\\Cnought$-semigroups on $\\BanachRaum$.\n Further let\n $(T^{(\\alpha)}_{i})_{\\alpha \\in \\Lambda_{i}}$\n be a net of expectation-approximants for $T_{i}$\n with associated distribution semigroups\n $(\\Gamma^{(\\alpha)}_{i}(t))_{t\\in\\realsNonNeg,\\alpha \\in \\Lambda_{i}}$\n for each $i \\in \\{1,2,\\ldots,d\\}$.\n Then for\n $\\boldsymbol{\\alpha}\\in\\prod_{i=1}^{d}\\Lambda_{i}$,\n and\n $\\boldsymbol{t} \\in \\realsNonNeg^{d}$\n it holds that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:yosida-stoch:prod:sig:article-stochastic-raj-dahya}\n \\displaystyle\n \\prod_{i=1}^{d}\n T^{(\\alpha_{i})}_{i}(t_{i})\n = \\Expected\\Big[\n \\displaystyle\n \\prod_{i=1}^{d}\n T_{i}(\\theta_{i})\n \\Big],\n \\end{eqnarray}\n\n \\noindent\n for any (equivalently: every) family of\n independent \\randomvar's\n ${\\theta_{i} \\distributedAs \\Gamma^{(\\alpha_{i})}_{i}(t)}$,\n $i\\in\\{1,2,\\ldots,d\\}$.\n\\end{prop}\n\n \\begin{proof}\n By definition of expectation-approximants we have\n $\n T^{(\\alpha_{i})}_{i}(t_{i})\n = \\Expected[\n T_{i}(\\theta_{i})\n ]\n $\n for each $i\\in\\{1,2,\\ldots,d\\}$\n and thus by independence\n $\n \\prod_{i=1}^{d}\n T^{(\\alpha_{i})}_{i}(t_{i})\n = \\prod_{i=1}^{d}\n \\Expected[T_{i}(\\theta_{i})]\n = \\Expected[\n \\prod_{i=1}^{d}\n T_{i}(\\theta_{i})\n ]\n $.\n \\end{proof}\n\nRestricting to the context of semigroups over Hilbert spaces yields a result,\nwhich can be utilised with regular polynomial evaluations.\nFor $p \\in \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}]$\nlet the \\highlightTerm{absolute degree of $p$}\ndenote the largest $n\\in\\naturalsZero$,\nsuch that $X_{i}^{n}$ or $X_{i}^{-n}$ occurs in some monomial in $p$,\nor else $0$ if $p=0$.\nIn particular, the absolute degree of $p$ is at most $1$\nif and only if\nthe only powers of the $X_{i}$ that occur in monomials in $p$ are $\\pm 1$.\n\n\\begin{prop\n\\makelabel{prop:expectation-approximants:poly:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\HilbertRaum$ be a Hilbert space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$\n be a commuting family of contractive $\\Cnought$-semigroups on $\\HilbertRaum$.\n Further let\n $(T^{(\\alpha)}_{i})_{\\alpha\\in\\Lambda_{i}}$\n be a net of expectation-approximants for $T_{i}$\n with associated distribution semigroups\n $(\\Gamma^{(\\alpha)}_{i})_{\\alpha\\in\\Lambda_{i}}$\n for each $i \\in \\{1,2,\\ldots,d\\}$.\n Then for\n $\\boldsymbol{\\alpha}\\in\\prod_{i=1}^{d}\\Lambda_{i}$,\n $\\boldsymbol{t} \\in \\realsNonNeg^{d}$\n and any regular polynomial\n $p \\in \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}]$\n with absolute degree at most $1$\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:yosida-stoch:poly:sig:article-stochastic-raj-dahya}\n p(\n T^{(\\lambda_{1})}_{1}(t_{1}),\n T^{(\\lambda_{2})}_{2}(t_{2}),\n \\ldots,\n T^{(\\lambda_{d})}_{d}(t_{d})\n )\n = \\Expected[\n p(\n T_{1}(\\theta_{1}),\n T_{2}(\\theta_{2}),\n \\ldots,\n T_{d}(\\theta_{d})\n )\n ]\n \\end{eqnarray}\n\n \\noindent\n for any (equivalently: every) family of\n independent \\randomvar's\n ${\\theta_{i} \\distributedAs \\Gamma^{(\\alpha_{i})}_{i}(t_{i})}$,\n $i\\in\\{1,2,\\ldots,d\\}$.\n\\end{prop}\n\n \\begin{proof}\n By linearity, it suffices to simply consider monomials of the form\n $p=\\prod_{i=1}^{n}X_{i}^{n_{i}}$\n where $\\mathbf{n} \\in \\{0, \\pm 1\\}^{d}$.\n Let\n $C_{1} \\colonequals \\mathop{\\textup{supp}}(\\mathbf{n}^{-})$,\n $C_{2} \\colonequals \\mathop{\\textup{supp}}(\\mathbf{n}^{+})$,\n and\n $K \\colonequals \\mathop{\\textup{supp}}(\\mathbf{n}) \\subseteq \\{1,2,\\ldots,d\\}$,\n where\n $\\mathbf{n}^{-} \\colonequals (n_{i}^{-})_{i=1}^{d}$\n and\n $\\mathbf{n}^{+} \\colonequals (n_{i}^{+})_{i=1}^{d}$.\n Then $\\isPartition{(C_{1},C_{2})}{K}$.\n By taking regular polynomial evaluations, we thus have to prove that\n\n \\vspace{-3\\parskip}\n $$\n \\prod_{i \\in C_{1}}T^{(\\lambda_{i})}_{i}(t_{i})^{\\ast}\n \\cdot\n \\prod_{j \\in C_{2}}T^{(\\lambda_{j})}_{j}(t_{j})\n =\n \\Expected\\Big[\n \\prod_{i \\in C_{1}}T_{i}(\\theta_{i})^{\\ast}\n \\cdot\n \\prod_{j \\in C_{2}}T_{j}(\\theta_{j})\n \\Big]\n $$\n\n \\noindent\n for any (equivalently: every) family of\n independent \\randomvar's\n ${\\theta_{i} \\distributedAs \\Gamma^{(\\alpha_{i})}_{i}(t)}$,\n $i \\in K$.\n This follows by applying\n \\Cref{%\n prop:basic-operations:expectation-approximations:sig:article-stochastic-raj-dahya,%\n prop:expectation-approximants:product:sig:article-stochastic-raj-dahya,%\n }.\n \\end{proof}\n\n\\begin{rem}\n For this paper we only need the statement in\n \\Cref{prop:expectation-approximants:poly:sig:article-stochastic-raj-dahya}\n to hold for regular polynomials of absolute degree at most $1$.\n If we nonetheless sought to widen the scope of this result,\n observe that limitations arise for powers $n$ of $X_{i}$ with $\\abs{n}\\geq 2$,\n since then the application of\n \\Crefit{prop:basic-operations:expectation-approximations:sig:article-stochastic-raj-dahya}{3}\n leads to random variables with different parameterisations.\n\\end{rem}\n\nAs an immediate consequence of \\Cref{prop:expectation-approximants:poly:sig:article-stochastic-raj-dahya},\nwe obtain:\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Transfer result]\n\\makelabel{lemm:positivity-transfer:poly:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$,\n $\\HilbertRaum$ be a Hilbert space,\n and\n $\\{T_{i}\\}_{i=1}^{d}$\n be a commuting family of contractive $\\Cnought$-semigroups on $\\HilbertRaum$.\n Further for each $i \\in \\{1,2,\\ldots,d\\}$ let\n $(T^{(\\alpha)}_{i})_{\\alpha \\in \\Lambda_{i}}$\n be a net of expectation-approximants for $T_{i}$.\n Then for any regular polynomial\n $p \\in \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}]$\n with absolute degree at most $1$,\n if\n $p(T_{1}(t_{1}),T_{1}(t_{2}),\\ldots,T_{d}(t_{d}))$\n is a positive operator\n for all $\\mathbf{t} \\in \\realsNonNeg^{d}$,\n then\n $p(T^{(\\alpha_{1})}_{1}(t_{1}),T^{(\\alpha_{1})}_{1}(t_{2}),\\ldots,T^{(\\alpha_{d})}_{d}(t_{d}))$\n is positive\n for all\n $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$\n and\n $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[First main result]{First main result: Expectation-approximants and polynomial bounds}\n\\label{sec:first-results:sig:article-stochastic-raj-dahya}\n\n\n\\noindent\nWe can now prove \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}.\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\n\\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n First observe by\n \\Cref{prop:expectation-approximants:basic:commuting-contractive-families:sig:article-stochastic-raj-dahya}\n that the expectation-approximants constitute commuting families of $\\Cnought$-semigroups,\n which converge \\wrt the \\topSOT-topology uniformly on compact subsets of $\\realsNonNeg^{d}$\n to the original family $\\{T_{i}\\}_{i=1}^{d}$.\n In particular, the implication\n \\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n holds by \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}.\n The implications\n \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\n hold by \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya},\n since these do not require the assumption of bounded generators.\n It remains to prove \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}.\n We first prove this under the assumption\n that the expectation-approximants have bounded generators.\n\n \\paragraph{\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}, under boundedness assumption on approximants:}\n Let\n $\n \\boldsymbol{\\alpha}\n \\in \\prod_{i=1}^{d}\\Lambda_{i}\n $\n be arbitrary.\n Let $K \\subseteq \\{1,2,\\ldots,d\\}$ be arbitrary.\n By assumption,\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n p_{K}(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))\n = \\displaystyle\n \\sum_{\\mathclap{\n \\isPartition{(C_{1},C_{2})}{K}\n }}\n \\displaystyle\n \\prod_{i \\in C_{1}}T_{i}(t_{i})^{\\ast}\n \\displaystyle\n \\prod_{j \\in C_{2}}T_{j}(t_{j})\n \\geq \\zeromatrix\n \\end{eqnarray*}\n\n \\noindent\n for all\n $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n By \\Cref{lemm:positivity-transfer:poly:sig:article-stochastic-raj-dahya},\n this positivity can be transferred to the family of expectation-approximants,\n \\idest\n ${\n p_{K}(T^{(\\alpha_{1})}_{1}(t_{1}),T^{(\\alpha_{2})}_{2}(t_{2}),\\ldots,T^{(\\alpha_{d})}_{d}(t_{d}))\n \\geq \\zeromatrix\n }$\n for all\n $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n Since this holds for all $K \\subseteq \\{1,2,\\ldots,n\\}$\n and since by assumption the semigroups in\n $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n have bounded generators,\n by \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n has a simultaneous regular unitary dilation.\n\n \n We have thus established the equivalence of\n \\punktcref{1}, \\punktcref{2}, \\punktcref{3}, \\punktcref{4}\n under the assumption that the expectation-approximants have bounded generators.\n Since by \\Cref{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya}\n one can always use the Hille- or Yosida-approximants (which by construction have bounded generators),\n the equivalences\n \\punktcref{1}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{3}\n hold in general. (\\textdagger)\n\n \\paragraph{\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}, without boundedness assumption on approximants:}\n By the above,\n \\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\n continue to hold.\n Exactly as argued above, \\punktcref{3} implies that\n ${\n p_{K}(T^{(\\alpha_{1})}_{1}(t_{1}),T^{(\\alpha_{2})}_{2}(t_{2}),\\ldots,T^{(\\alpha_{d})}_{d}(t_{d}))\n \\geq \\zeromatrix\n }$\n for all\n $\\mathbf{t} \\in \\realsNonNeg^{d}$,\n $K\\subseteq\\{1,2,\\ldots,d\\}$,\n and\n $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$.\n Let $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$ be arbitrary.\n By the general validity of \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1} (see (\\textdagger))\n applied to $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$,\n it follows that\n $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n has a simultaneous regular unitary dilation.\n Returning to the current context, this means that \\punktcref{4} holds.\n\\end{proof}\n\n\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:explicit-requirement-contractive-not-needed:sig:article-stochastic-raj-dahya}\n In \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\n since \\punktcref{1}, \\punktcref{2}, and \\punktcref{3}\n each imply that the $T_{i}$ are contractive,\n for the equivalences\n \\punktcref{1}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{3}\n it is not necessary to explicitly demand that the $T_{i}$ are contractive.\n To see that \\punktcref{2} implies that the $T_{i}$ are contractive,\n consider polynomial bounds applied to the regular polynomials\n $X_{i}$ for each $i\\in\\{1,2,\\ldots,d\\}$.\n To see that \\punktcref{3} implies that the $T_{i}$ are contractive,\n let $i\\in\\{1,2,\\ldots,d\\}$ be arbitrary.\n By considering the polynomial, $p_{\\{i\\}}$,\n we have that\n $(\\onematrix-T_{i}(t)^{\\ast}) + (\\onematrix-T_{i}(t)) \\geq \\zeromatrix$\n for all $t\\in\\realsNonNeg$.\n It readily follows that the generator, $A_{i}$, of $T_{i}$\n is dissipative and thus that $T_{i}$ is contractive.\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:significance-of-result-other-dilation:sig:article-stochastic-raj-dahya}\n If we replace \\emph{\\usesinglequotes{simultaneous regular unitary dilations}}\n by \\emph{\\usesinglequotes{simultaneous unitary dilations}},\n classifications via bounds of algebraic expressions\n include \\cite[Theorem~2.2]{LeMerdy1996DilMultiParam} in the continuous setting,\n and \\cite[Corollaries~4.9]{Pisier2001bookCBmaps} in the discrete setting (\\idest for tuples of commuting contractions).\n In the discrete setting,\n the existence of a simultaneous \\emph{power dilation}\n is characterised by the \\emph{complete boundedness}\n of polynomials defined on the operators.\n In the continuous setting,\n le~Merdy characterised the existence of a simultaneous unitary dilation\n by the complete boundedness of a certain functional calculus map\n generated by the resolvents of the semigroups\n and defined on an algebra of holomorphic functions.\n Adding to this picture, our result in\n \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n (see also \\Cref{rem:explicit-requirement-contractive-not-needed:sig:article-stochastic-raj-dahya})\n answers \\Cref{qstn:polynomial:sig:article-stochastic-raj-dahya} positively\n and we now know in full generality\n that finite commuting families of $\\Cnought$-semigroups over Hilbert spaces\n have a simultaneous regular unitary dilation if and only if\n they satisfy regular polynomial bounds.\n (In \\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya} a further characterisation\n of regular unitary dilations via the complete positivity of a functional calculus shall be presented,\n which is more general than the regular polynomial bounds and further adds to this picture.)\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:theorem-positivity-in-a-neighbourhood:sig:article-stochastic-raj-dahya}\n Consider a neighbourhood $U \\subseteq \\realsNonNeg^{d}$ of $\\zerovector$\n and let (\\punktref{3}') be the assertion of the positivity of the operators in \\punktcref{3}\n for $\\mathbf{t} \\in U$ instead of for all $\\mathbf{t}\\in\\realsNonNeg^{d}$.\n We show that \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n holds with \\punktcref{3} replaced by (\\punktref{3}').\n For this it suffices to show that (\\punktref{3}') implies \\punktcref{1}\n (\\idest that $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation).\n\n To this end, fix some ${a\\in\\realsPos}$ such that ${U \\supseteq [0,\\:a)^{d}}$\n and consider the subnet of Hille-approximants:\n $(T^{(\\lambda)}_{i})_{\\lambda\\in(a^{-1},\\:\\infty)}$ for $T_{i}$\n for each ${i\\in\\{1,2,\\ldots,d\\}}$.\n Clearly, taking subnets does not affect the fact that these are expectation-approximants.\n Thus applying \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n yields that $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation,\n if and only if\n $\\{T^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$\n has a simultaneous regular unitary dilation\n for each ${\\boldsymbol{\\lambda}\\in(a^{-1},\\:\\infty)^{d}}$.\n Since the Hille-approximants have bounded generators,\n by \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n this holds if and only if the generators\n $\\{A^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$\n of the Hille-approximants are completely dissipative\n for each ${\\boldsymbol{\\lambda}\\in(a^{-1},\\:\\infty)^{d}}$.\n By construction of the Hille-approximants, this holds if and only if\n\n \\vspace{-3\\parskip}\n $$\n (-\\tfrac{1}{2})^{\\card{K}}\n \\sum_{\\isPartition{(C_{1},C_{2})}{K}}\n \\Big(\n \\prod_{i\\in C_{1}}\n \\Big(\n \\lambda_{i}\n (T_{i}(\\tfrac{1}{\\lambda_{i}})-\\onematrix)\n \\Big)\n \\Big)^{\\ast}\n \\prod_{j\\in C_{2}}\n \\Big(\n \\lambda_{j}\n (T_{j}(\\tfrac{1}{\\lambda_{j}})-\\onematrix)\n \\Big)\n \\geq \\zeromatrix\n $$\n\n \\noindent\n for all ${\\boldsymbol{\\lambda}\\in(a^{-1},\\:\\infty)^{d}}$\n and all $K \\subseteq \\{1,2,\\ldots,d\\}$.\n This holds if and only if\n ${p_{K}(T_{1}(t_{1}), T_{1}(t_{2}),\\ldots,T_{d}(t_{d})) \\geq \\zeromatrix}$\n for all ${\\mathbf{t}\\in(0,\\:a)^{d}}$\n and all $K \\subseteq \\{1,2,\\ldots,d\\}$.\n This in turn is clearly implied by (\\punktref{3}').\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:other-approximants:sig:article-stochastic-raj-dahya}\n By\n \\punktcref{1}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{4}\n of\n \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n as well as\n \\Cref{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya},\n we have answered\n \\Cref{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}\n positively\n for a large class of naturally definable examples:\n The simultaneous regular unitary dilatability\n of a commuting contractive family\n is characterised by\n the simultaneous regular unitary dilatability\n of families of semigroups\n in any given net of expectation-approximants,\n \\exempli the nets of Hille- and Yosida-approximants.\n It would be interesting to know whether this characterisation holds\n for other classically defined approximants,\n such as the approximants that occur in Kendall's formula\n and the semigroup version of the Post-Widder theorem\n (\\cf\n \\cite[Theorem~10.4.3~and~11.6.6]{Hillephillips1957faAndSg},\n \\cite[Theorems~2--3]{Chung1962exp}%\n ).\n In these cases, in place of the stochastic methods used in the present paper,\n other techniques such as product formulae used with path integrals,\n \\exempli Chernoff approximations\n (see \\cite{Chernoff1968article,Chernoff1974book,Butko2020chernoff}),\n may be better suited.\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:complete-dissipativity:sig:article-stochastic-raj-dahya}\n Let $\\{A_{i}\\}_{i=1}^{d}$ be the generators of a commuting family,\n $\\{T_{i}\\}_{i=1}^{d}$, of $\\Cnought$-semigroups on a Hilbert space $\\HilbertRaum$.\n Consider now the subset\n $D \\subseteq \\HilbertRaum$,\n of elements $\\xi$ which lie in the domain of\n $A_{k_{n}} \\cdot \\ldots \\cdot A_{k_{2}} \\cdot A_{k_{1}}$\n and such that the value of\n ${(A_{k_{n}} \\cdot \\ldots \\cdot A_{k_{2}} \\cdot A_{k_{1}})\\xi}$\n does not depend on the order of the $k_{i}$\n for injective sequences\n $(k_{i})_{i=1}^{n} \\subseteq \\{1,2,\\ldots,d\\}$,\n $n\\in\\{1,2,\\ldots,d\\}$.\n It can be shown that $D$ is a dense linear subspace of $\\HilbertRaum$\n (see \\Cref{prop:dense-subspace-multi-param:sig:article-stochastic-raj-dahya}).\n One may thus extend the notion of complete dissipativity to\n families of generators $\\{A_{i}\\}_{i=1}^{d}$\n (without the boundedness assumption),\n by demanding that\n\n \\vspace{-3\\parskip}\n $$\n (-\\tfrac{1}{2})^{\\card{K}}\n \\sum_{\\isPartition{(C_{1},C_{2})}{K}}\n \\brktLong{\n \\Big(\n \\prod_{j\\in C_{2}}\n A_{j}\n \\Big)\n \\xi\n }{\n \\Big(\n \\prod_{i\\in C_{1}}\n A_{i}\n \\Big)\n \\xi\n }\n \\geq 0\n $$\n\n \\noindent\n for all ${\\xi \\in D}$ and ${K \\subseteq \\{1,2,\\ldots,d\\}}$.\n\n Appealing to condition \\eqcref{it:3:thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n of \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\n if $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation,\n then by taking limits of the positive expressions\n $\n \\frac{1}{2^{K}\\prod_{i=1}^{d}t_{i}}\n \\brktLong{\n p_{K}(\n T_{1}(t_{1}),\n T_{2}(t_{2}),\n \\ldots,\n T_{d}(t_{d})\n )\\xi\n }{\\xi}\n $\n for ${\\realsPos \\ni t_{i} \\longrightarrow 0}$\n successively for each ${i \\in \\{1,2,\\ldots,d\\}}$\n and for each ${\\xi \\in D}$,\n we obtain that $\\{A_{i}\\}_{i=1}^{d}$ is completely dissipative\n by the above definition.\n However, it is unclear whether the reverse implication holds.\n The approach used in \\cite[Theorem~1.1]{Dahya2023dilation}\n to link complete dissipativity to\n a previously known condition%\n \\footnote{\n \\emph{Brehmer positivity},\n see\n \\cite[Theorem~3.2]{Ptak1985}.\n }\n which characterises the existence of simultaneous regular unitary dilations in general,\n relies on asymptotic expressions,\n which in turn rely on the boundedness of the generators.\n Hence an alternative approach is needed for the unbounded setting.\n It would thus be of interest to know whether the above (or an alternative)\n definition of complete dissipativity can be shown to be equivalent\n to any (and thus all) of the conditions in \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}.\n\\end{rem}\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Functional calculi associated with dilations]{Functional calculi associated with dilations}\n\\label{sec:functional-calculus:sig:article-stochastic-raj-dahya}\n\n\\noindent\nWe now leave the setting of commuting families and turn our attention\nto classical dynamical systems modelled by\n\\topSOT-continuous homomorphisms between topological monoids\nand bounded operators over a Hilbert space.\nIn this section we provide characterisations of unitary and regular unitary dilations\nvia \\emph{functional calculi} defined on (subalgebras of) certain $C^{\\ast}$-algebras\nrelated to topological groups.\nWe thus begin in \\S{}\\ref{sec:functional-calculus:harmonic-analysis:sig:article-stochastic-raj-dahya}\nby recalling the correspondence between ${}^{\\ast}$-representations of group $C^{\\ast}$-algebras\nand unitary representations of topological groups.\nThen in \\S{}\\ref{sec:functional-calculus:continuous:sig:article-stochastic-raj-dahya}\nwe shall use the Wittstock-Haagerup result,\nwhich involves extending and then dilating \\emph{completely bounded} maps.\nWe build on this to generalise le~Merdy's approach to characterise unitary dilatability.\nFinally, in \\S{}\\ref{sec:functional-calculus:discrete:sig:article-stochastic-raj-dahya}\nwe shall use Averson's result and Stinespring's theorem,\nwhich involves extending and then dilating \\emph{completely positive} maps.\nWe build on this to obtain a characterisation of regular unitary dilations\nsimilar to that of Sz.-Nagy and Foias\n(see \\cite[Theorem~7.1~b]{Nagy1970}).\nThe underlying algebras involved in the two functional calculi\ndemonstrate that unitary dilations are in some sense a \\emph{continuous} phenomenon,\nwhilst regular unitary dilations are in some sense a \\emph{discrete} phenomenon.\n\nThroughout this section, $(G,\\cdot,e)$ (or simply: $G$)\nshall denote a locally compact topological group%\n\\footnote{\n Since we are only concerned with continuous maps between $G$\n and other Hausdorff topological groups\n (\\exempli the group of unitaries on a Hilbert space under the $\\topSOT$-topology),\n it is not important to assume that $G$ be Hausdorff\n (\\cf \\cite[\\S{}1.2]{Deitmar2014bookHarmonicAn}).\n Nonetheless, all of our examples\n (see \\Cref{sec:introduction:monoids:sig:article-stochastic-raj-dahya})\n are Hausdorff.\n}\nand $M$ shall denote a (closed) submonoid of $G$\nso that\n $(M,\\cdot,e)$ (or simply $M$),\nequipped with the relative topology,\ncomprises a (locally compact) topological monoid.\nWe furthermore let $\\lambda_{G}$ denote a left-invariant Haar-measure on $G$\nand express integrals of $G$ via ${\\int_{x \\in G}\\:\\cdot\\:\\dee x}$.\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Abstract harmonic analysis]{Abstract harmonic analysis}\n\\label{sec:functional-calculus:harmonic-analysis:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn order to study dilations of classical dynamical systems on Hilbert spaces,\nwe shall make use of a fundamental relationship between unitary representations\nand ${}^{\\ast}$-representations of $C^{\\ast}$-algebras.\nWe recall these facts here.\nFor a detailed exposition,\nsee \\exempli\n \\cite[\\S{}3.2 and \\S{}7.1]{Folland2015bookHarmonicAnalysis},\n \\cite[\\S{}3.3]{Deitmar2014bookHarmonicAn}.\nLet $G$ be a locally compact group\n(for which we can fix a left-invariant Haar measure)\nand let ${\\Delta(\\cdot) \\colon G\\to(\\realsPos,\\cdot,1)}$\nbe the \\emph{modular function},\nwhich is a continuous homomorphism.\nThen $L^{1}(G)$ forms a Banach ${}^{\\ast}$-algebra\nunder the convolution operation (viewed as \\usesinglequotes{multiplication})\nand involution defined by\n\n \\vspace{-3\\parskip}\n $$\n (f_{1}\\ast f_{2})(x)\n \\colonequals\n \\int_{y \\in G}\n f_{1}(y)f_{2}(y^{-1}x)\n \\:\\dee y\n $$\n\n\\noindent\nand\n\n \\vspace{-3\\parskip}\n $$\n f^{\\ast}(x)\n \\colonequals\n (f(x^{-1}))^{\\ast}\n \\Delta(x^{-1})\n $$\n\n\\noindent\nrespectively,\nfor $f, f_{1},f_{2}\\in L^{1}(G)$, $x\\in G$.\nIt can then be shown, for any Hilbert space $\\HilbertRaum$,\nthat there is a natural a correspondence between\n non-degenerate%\n\\footnote{\n \\idest there is no $\\xi\\in\\HilbertRaum\\mathbin{\\setminus}\\{\\zerovector\\}$\n such that\n $\\pi(f)\\xi=\\zerovector$\n for all $f \\in L^{1}(G)$.\n This is the case, \\exempli for irreducible representations.\n}\n ${}^{\\ast}$-representations, $\\pi$, of $(L^{1}(G),\\ast,{}^{\\ast})$\n on $\\HilbertRaum$\nand \\topSOT-continuous unitary representations, $U$, of $G$\n on $\\HilbertRaum$\ngiven by\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:correspondence-unitary-to-star-repr:shift:sig:article-stochastic-raj-dahya}\n U(x)\\pi(f) = \\pi(L_{x}f) = \\pi(f(x^{-1}\\cdot ))\n \\end{eqnarray}\n\n\\noindent\nand\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n \\brkt{\\pi(f)\\xi}{\\eta}\n = \\int_{x \\in G}\n f(x)\\brkt{U(x)\\xi}{\\eta}\n \\:\\dee g\n \\end{eqnarray}\n\n\\noindent\nfor $f\\in L^{1}(G)$, $x\\in G$, $\\xi,\\eta\\in\\HilbertRaum$.\n\nLet $\\hat{G}$ denote the set of all\n irreducible ${}^{\\ast}$-representations of $L^{1}(G)$\n (equivalently: irreducible unitary representations of $G$),\n up to unitary equivalence.\nOne can then equip\n $(L^{1}(G),\\ast,{}^{\\ast})$\nwith the norm\n\n \\vspace{-3\\parskip}\n $$\n \\norm{f}_{\\ast}\n \\colonequals\n \\sup_{[\\pi]\\in\\hat{G}}\n \\norm{\\pi(f)}\n $$\n\n\\noindent\nfor $f\\in L^{1}(G)$.\nThis renders $(L^{1}(G),\\ast,{}^{\\ast})$\na dense ${}^{\\ast}$-subalgebra of a $C^{\\ast}$-algebra,\nwhich is referred to as the\n\\highlightTerm{group $C^{\\ast}$-algebra} for $G$,\nand is denoted $C^{\\ast}(G)$.\nNote that in particular $\\norm{f}_{\\ast} \\leq \\norm{f}$\nholds for all $f \\in L^{1}(G)$.\n\n\\begin{conv}\n For simplicity we shall view $L^{1}(G)$ as a subset of $C^{\\ast}(G)$.\n We shall interchangeably denote the group $C^{\\ast}$-algebra for $G$\n as $C^{\\ast}(G)$ and as $\\quer{L^{1}(G)}$.\n Equipping $G$ with the discrete topology yields a locally compact group\n with the counting measure as the Haar measure.\n In this case, the $L^{1}$-space is simply $\\ell^{1}(G)$.\n We shall thus denote the group $C^{\\ast}$-algebra associated\n with the discretised $G$ via $\\quer{\\ell^{1}(G)}$.\n\\end{conv}\n\nAs a $C^{\\ast}$-algebra, $C^{\\ast}(G)$ has a unique unital extension.\nNote that $L^{1}(G)$ contains a unital element\nif and only if $G$ is discrete,\nin which case, the unit is $\\delta_{e}$.\nFor this reason,\nwe shall denote the unital extension of $C^{\\ast}(G)$ by\n\n \\vspace{-3\\parskip}\n $$\n \\complex\\cdot\\delta_{e} + C^{\\ast}(G)\n $$\n\n\\noindent\nin case $G$ is continuous.\nIf $G$ is discrete, we have that\n $\\complex\\cdot\\delta_{e} + C^{\\ast}(G) = C^{\\ast}(G)$.\nOtherwise the above extension can be understood as\n $\\complex\\cdot\\delta_{e} \\oplus C^{\\ast}(G)$.\n\nRelying on the density of $L^{1}(G)$ in $C^{\\ast}(G)$,\nthe above correspondence can be restated,\nthe proof of which is due to Folland.\n\n\\begin{prop}[Correspondence between ${}^{\\ast}$- and unitary representations]\n\\makelabel{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}\n Let\n $G$ be a locally compact topological group\n and\n $\\HilbertRaum$ a Hilbert space.\n Then for every\n \\topSOT-continuous unitary representation, $U$, of\n $G$ on $\\HilbertRaum$\n there exists a\n non-degenerate ${}^{\\ast}$-representation, $\\pi = \\pi_{U}$, of\n $C^{\\ast}(G)$ on $\\HilbertRaum$,\n such that\n \\eqcref{eq:correspondence-unitary-to-star-repr:shift:sig:article-stochastic-raj-dahya}\n and\n \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n hold.\n And for every\n non-degenerate ${}^{\\ast}$-representation, $\\pi$, of\n $C^{\\ast}(G)$ on $\\HilbertRaum$,\n there exists an\n \\topSOT-continuous unitary representation, $U = U_{\\pi}$, of\n $G$ on $\\HilbertRaum$,\n such that\n \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n holds.\n The constructions\n ${U\\mapsto\\pi_{U}}$ and ${\\pi\\mapsto U_{\\pi}}$\n establish a correspondence.\n between \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$\n and non-degenerate ${}^{\\ast}$-representations of $C^{\\ast}(G)$ on $\\HilbertRaum$.\n\\end{prop}\n\n \\begin{proof}\n For the first claim, by\n \\cite[Theorem~3.9]{Folland2015bookHarmonicAnalysis},\n there exists a non-degenerate ${}^{\\ast}$-representation, $\\pi$, of\n $L^{1}(G)$ on $\\HilbertRaum$\n satisfying\n \\eqcref{eq:correspondence-unitary-to-star-repr:shift:sig:article-stochastic-raj-dahya}\n and\n \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}.\n Using Zorn's lemma, one can decompose\n $\\HilbertRaum$\n into closed $\\pi$-invariant subspaces with cyclic vectors,\n $\\HilbertRaum=\\bigoplus_{i}\\HilbertRaum_{i}$,\n thereby obtaining irreducible ${}^{\\ast}$-representations, $\\pi_{i}$,\n of $L^{1}(G)$ on each $\\HilbertRaum_{i}$.\n By construction of the $\\norm{\\cdot}_{\\ast}$-norm,\n each $\\pi_{i}$ and thus $\\pi$ itself are contractive.\n It follows that $\\pi$ can be extended\n to a bounded linear operator between\n $\\quer{L^{1}(G)} = C^{\\ast}(G)$\n and $\\BoundedOps{\\HilbertRaum}$,\n which we may also call $\\pi$.\n Since the algebraic operations in $C^{\\ast}(G)$ are continuous,\n $\\pi$ remains a ${}^{\\ast}$-representation.\n\n Towards the second claim, by density, $\\pi$ is a non-degenerate ${}^{\\ast}$-representation\n of $L^{1}(G)$ on $\\HilbertRaum$,\n and thus by \\cite[Theorem~3.11]{Folland2015bookHarmonicAnalysis},\n there exists an \\topSOT-continuous unitary representation, $U$, of\n $G$ on $\\HilbertRaum$,\n satisfying\n \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}.\n\n Towards the final claim, we first show that $\\pi_{U_{\\pi}} = \\pi$\n for each ${}^{\\ast}$-representation, $\\pi$, of $C^{\\ast}(G)$ on $\\HilbertRaum$.\n Using\n \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n yields\n $\n \\brkt{\\pi_{U_{\\pi}}(f)xi}{\\eta}\n = \\int_{x \\in G}f(x)\\brkt{U_{\\pi}(x)\\xi}{\\eta}\\:\\dee x\n = \\brkt{\\pi(f)\\xi}{\\eta}\n $\n for all $f\\in L^{1}(G)$ and $\\xi,\\eta\\in\\HilbertRaum$.\n By the density of $L^{1}(G)$ in $C^{\\ast}(G)$\n and continuity of $\\pi$, $\\pi_{U_{\\pi}}$,\n it follows that $\\pi_{U_{\\pi}} = \\pi$.\n To show that $U_{\\pi_{U}} = U$\n for each \\topSOT-continuous unitary representation, $U$, of\n $G$ on $\\HilbertRaum$,\n using\n \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n yields\n $\n \\int_{K}\\brkt{U_{\\pi_{U}}(x)\\xi}{\\eta}\\:\\dee x\n = \\brkt{\\pi_{U}(\\einser_{K})xi}{\\eta}\n = \\int_{K}\\brkt{U(x)\\xi}{\\eta}\\:\\dee x\n $\n for all compact $K \\subseteq G$\n and all $\\xi,\\eta\\in\\HilbertRaum$.\n By the \\topWOT-continuity of $U$, $U_{\\pi_{U}}$,\n it follows that $U_{\\pi_{U}} = U$.\n \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[The continuous functional calculus]{The continuous functional calculus}\n\\label{sec:functional-calculus:continuous:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor a locally compact topological group $G$ and closed submonoid $M \\subseteq G$,\nwe consider\n\n \\vspace{-3\\parskip}\n $$\n \\LOneCPlus{M}{G}\n \\colonequals\n \\{\n f \\in L^{1}(G)\n \\mid\n \\quer{\\mathop{\\textup{supp}}}(f) \\subseteq M,\n ~\\text{compact}\n \\},\n $$\n\n\\noindent\nwhich is a subalgebra of the convolution algebra $(L^{1}(G),\\ast)$\nand thus of the unital $C^{\\ast}$-algebra ${\\complex\\cdot\\delta_{e} + C^{\\ast}(G)}$.\nFor an \\topSOT-continuous homomorphism,\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$,\non $\\HilbertRaum$, consider the map\n ${\\funcCalcCts \\colon \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G} \\to \\BoundedOps{\\HilbertRaum}}$\ndefined by\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:functional-calc:cts:sig:article-stochastic-raj-dahya}\n \\funcCalcCts(c\\delta_{e} + f)\n = c\\onematrix\n + \\sotInt_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)T(x)\n \\:\\dee x\n \\end{eqnarray}\n\n\\noindent\nfor $f\\in\\LOneCPlus{M}{G}$, $c\\in\\complex$.\nIt is easy to verify that $\\funcCalcCts$ is a linear unital map\nwhich satisfies\n $\n \\funcCalcCts(f \\ast g)\n = \\funcCalcCts(f)\n \\funcCalcCts(g)\n $\nfor $f,g\\in\\LOneCPlus{M}{G}$.\nWe shall refer to this unital homomorphism\nas the \\highlightTerm{continuous functional calculus}\nassociated with $T$.\n\n\\begin{rem}\n\\makelabel{rem:continuous-functional-calculus-le-merdy-phillips:sig:article-stochastic-raj-dahya}\n Consider the case $(G,M)=(\\reals^{d},\\realsNonNeg^{d})$.\n Let\n $\\HilbertRaum$ be a Hilbert space\n and\n ${T \\colon \\realsNonNeg^{d} \\to \\BoundedOps{\\HilbertRaum}}$\n be an \\topSOT-continuous contractive homomorphism on $\\HilbertRaum$.\n Further let\n $\\{T_{i}\\}_{i=1}^{d}$\n be the corresponding commuting family of contractive $\\Cnought$-semigroups.\n Then \\eqcref{eq:functional-calc:cts:sig:article-stochastic-raj-dahya}\n can be naturally extended\n to\n ${\n \\mathcal{B}\n \\colonequals\n \\complex\\cdot\\delta_{\\zerovector} \\oplus \\mathcal{B}_{0}\n }$,\n where\n ${\n \\mathcal{B}_{0}\n \\colonequals\n \\{f\\in L^{1}(\\reals^{d}) \\mid \\mathop{\\textup{supp}}(f) \\subseteq \\realsNonNeg^{d}\\}\n }$.\n This extension of $\\funcCalcCts$ to $\\mathcal{B}$\n is essentially a restriction of the \\emph{Phillips calculus}\n (see \\exempli\n \\cite[Lemma~VIII.1.12~{[$\\ast$]}]{Dunfordschwartz1988BookLinOpI},\n \\cite[Proposition~3.3.5]{Reissig2005abstractresolvent}%\n )\n applied to the generators $\\{A_{i}\\}_{i=1}^{d}$ of $\\{T_{i}\\}_{i=1}^{d}$.\n Consider now arbitrary\n $\\mathbf{n}\\in\\naturalsPos^{d}$\n and\n $\\boldsymbol{\\lambda}\\in\\realsPos^{d}$,\n and let\n $f\\in \\mathcal{B}_{0}$\n be defined by\n ${\n f(\\mathbf{t})\n \\colonequals\n \\prod_{i=1}^{d}\n \\lambda_{i}\\frac{(\\lambda_{i}t_{i})^{n_{i}-1}}{(n-1)!}\n e^{-\\lambda_{i}t_{i}}\n }$\n for $\\mathbf{t}\\in\\realsNonNeg^{d}$.\n One has\n ${\n (\\Fourier f)(\\boldsymbol{\\omega})\n = \\prod_{i=1}^{d}\n (\\frac{\\lambda_{i}}{\\lambda_{i} + \\iunit\\omega_{i}})^{n_{i}}\n }$\n for $\\boldsymbol{\\omega}\\in\\reals^{d}$,\n where ${\\Fourier \\colon L^{1}(\\reals^{d}) \\to \\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}_{0}{\\reals^{d}}}$\n denotes the Fourier transform.\n Then\n ${\n \\funcCalcCts(f)\n = \\prod_{i=1}^{d}\n \\sotInt_{s=0}^{\\infty}\n \\lambda_{i}\\frac{(\\lambda_{i}s)^{n_{i}-1}}{(n-1)!}\n e^{-\\lambda_{i}s}\n T_{i}(s)\n \\:\\dee s\n = \\prod_{i=1}^{d}\n (\\lambda_{i} \\opResolvent{A_{i}}{\\lambda_{i}})^{n_{i}}\n }$.\n The Phillips calculus is thus a common extension\n of the continuous functional calculus\n and (upon application of the Fourier transform)\n the one defined by le~Merdy in \\cite[Definition~2.1]{LeMerdy1996DilMultiParam}.\n\\end{rem}\n\nThe main result in this subsection is thus inspired by le~Merdy's characterisation of simultaneous unitary dilations.\nBy simplifying his approach and utilising results from abstract harmonic analysis we obtain a generalisation.\nBefore presenting this, we need the following approximation.\n\n\\begin{prop}\n\\makelabel{prop:func-calc-cts:basic:shift-unit:sig:article-stochastic-raj-dahya}\n Let $G$ be a locally compact topological group and\n $M \\subseteq G$ a closed submonoid.\n Further let\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n be an \\topSOT-continuous homomorphism.\n If $M$ is $e$-joint,\n then there exists a net\n $(f_{i})_{i} \\subseteq \\LOneCPlus{M}{G}$\n such that\n $\\norm{f_{i}}_{\\ast} \\leq 1$ for all $i\\in I$,\n ${\\norm{f_{i} \\ast g - g}_{\\ast} \\underset{i}{\\longrightarrow} 0}$\n for all $g \\in L^{1}(G)$,\n and\n $\\funcCalcCts(f_{i}) \\underset{i}{\\longrightarrow} \\onematrix$\n \\wrt the \\topSOT-topology.\n\\end{prop}\n\n \\begin{proof}\n Let $\\mathcal{N}$ be the filter of all compact neighbourhoods of\n the group identity $e \\in G$.\n By compactness and $e$-jointedness,\n $0 < \\lambda_{G}(K \\cap M) \\leq \\lambda_{G}(K) < \\infty$\n for all $K \\in \\mathcal{N}$.\n Thus\n $f_{K} \\colonequals \\frac{1}{\\lambda_{G}(K \\cap M)}\\einser_{K \\cap M}$\n is a well-defined element of $L^{1}(G)$\n with $\\quer{\\mathop{\\textup{supp}}}(f) = K \\cap M \\subseteq M$\n for each $K\\in\\mathcal{N}$,\n and moreover\n $\n \\norm{f}_{\\ast}\n \\leq \\norm{f}_{1}\n = 1\n $.\n Hence it suffices to consider the net\n $(f_{K})_{K\\in\\mathcal{N}}$\n directly ordered by reverse inclusion.\n\n Let $g \\in L^{1}(G)$ be arbitrary.\n Then\n $\\norm{f_{K} \\ast g - g}_{\\ast} \\leq \\norm{f_{K} \\ast g - g}_{1}$\n for each $h\\in\\realsPos$.\n Since convolution is $L^{1}$-continuous\n (see \\exempli \\cite[Proposition~2.40a)]{Folland2015bookHarmonicAnalysis}) and $\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}_{c}{G}$ is dense in $L^{1}(G)$,\n it suffices to prove that\n ${\\norm{f_{K} \\ast g - g}_{1} \\underset{K}{\\longrightarrow} 0}$\n for each $g\\in\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}_{c}{G}$.\n This is a straightforward matter that can be derived\n using uniform continuity arguments.\n\n Towards the final claim, one has\n $\n \\norm{\\funcCalcCts(f_{K})\\xi - \\xi}\n = \\normLong{\n \\Big(\n \\frac{1}{\\lambda_{G}(K \\cap M)}\n \\sotInt_{x \\in K \\cap M}\n T(x)\n \\:\\dee x\n \\Big)\n \\xi\n -\n \\xi\n }\n \\leq\n \\frac{1}{\\lambda_{G}(K \\cap M)}\n \\int_{x \\in K \\cap M}\n \\norm{T(x)\\xi - T(e)\\xi}\n \\:\\dee x\n \\leq\n \\sup_{x \\in K \\cap M}\n \\norm{(T(x) - T(e))\\xi}\n $\n for each $\\xi\\in\\HilbertRaum$,\n whereby the latter expression\n converges to $0$ since $T$ is \\topSOT-continuous.\n \\end{proof}\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Generalisation of le~Merdy's characterisation of dilations]\n\\makelabel{lemm:cts-functional-calculus:sig:article-stochastic-raj-dahya}\n Let $G$ be a locally compact topological group and\n $M \\subseteq G$ an $e$-joint closed submonoid.\n Further let\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n be an \\topSOT-continuous homomorphism.\n Then $T$ has a unitary dilation if and only if $\\funcCalcCts$ is completely bounded\n with $\\normCb{\\funcCalcCts} \\leq 1$.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\n \\begin{proof}\n Via the GNS-construction, we may view\n the unital $C^{\\ast}$-algebra,\n ${\\mathcal{A} \\colonequals \\complex\\cdot\\delta_{e} + C^{\\ast}(G)}$\n as a unital $C^{\\ast}$-subalgebra of $\\BoundedOps{\\HilbertRaum_{0}}$\n for some Hilbert space $\\HilbertRaum_{0}$.\n\n \\paragraph{Necessity:}\n Suppose that $(\\HilbertRaum_{1},U,r)$ is a unitary dilation of $T$.\n By the correspondence between unitary representations and non-degenerate ${}^{\\ast}$-representations\n in abstract harmonic analysis\n (see \\Cref{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}),\n there exists a non-degenerate ${}^{\\ast}$-representation\n ${\n \\pi \\colon\n C^{\\ast}(G)\n \\to\n \\BoundedOps{\\HilbertRaum_{1}}\n }$\n such that \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya} holds.\n Now, $\\pi$ can be extended to\n ${\n \\tilde{\\pi} \\colon\n \\complex\\cdot\\delta_{e} + C^{\\ast}(G)\n \\to\n \\BoundedOps{\\HilbertRaum_{1}}\n }$\n defined by\n ${\\tilde{\\pi}(c\\delta_{e} + f) \\colonequals c\\onematrix + \\pi(f)}$\n for $f\\in C^{\\ast}(G)$ and $c\\in\\complex$.\n By non-degeneracy, $\\tilde{\\pi}$ is a well-defined%\n \\footnote{\n in particular, if $C^{\\ast}(G)$ already contains the $\\delta_{e}$,\n then by non-degeneracy $\\pi(\\delta_{e})=\\onematrix$\n must hold.\n }\n unital ${}^{\\ast}$-algebra representation.\n Using the unitary dilation and the ${}^{\\ast}$-representation,\n one obtains\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\\funcCalcCts(c\\delta_{e} + f)\\xi}{\\eta}\n &= &\\brktLong{\n \\Big(\n c\\onematrix\n +\n \\displaystyle\n \\sotInt_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)T(x)\n \\:\\dee x\n \\Big)\n \\xi\n }{\\eta}\\\\\n &= &c\\brkt{\\xi}{\\eta}\n +\n \\displaystyle\n \\int_{x\\in\\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)\n \\underbrace{\n \\brkt{T(x)\\xi}{\\eta}\n }_{=\\brkt{r^{\\ast}U(x)r\\xi}{\\eta}}\n \\:\\dee x\\\\\n &= &c\\brkt{r\\xi}{r\\eta}\n +\n \\displaystyle\n \\int_{x\\in\\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)\n \\brkt{U(x)r\\xi}{r\\eta}\n \\:\\dee x\\\\\n &\\eqcrefoverset{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}{=}\n &\\brkt{c\\onematrix\\,r\\xi}{r\\eta}\n +\n \\brkt{\\pi(f)r\\xi}{r\\eta}\\\\\n &= &\\brkt{r^{\\ast}\\tilde{\\pi}(c\\cdot\\delta_{e} + f)r\\xi}{\\eta}\n \\end{eqnarray*}\n\n \\noindent\n for all\n $f \\in \\LOneCPlus{M}{G} \\subseteq L^{1}(G)$,\n $c\\in\\complex$,\n $\\xi,\\eta\\in\\HilbertRaum$.\n Thus $\\funcCalcCts(a) = r^{\\ast}\\tilde{\\pi}(a)r$\n for all $a\\in \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}$.\n Let $n\\in\\naturals$ and $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(\\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G})$.\n Then\n $\n (\\funcCalcCts\\otimes\\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n = (\\funcCalcCts(a_{ij}))_{ij}\n = (r^{\\ast}\\tilde{\\pi}(a_{ij})r)_{ij}\n = (r \\otimes \\onematrix_{M_{n}})^{\\ast}\n (\\tilde{\\pi} \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n (r \\otimes \\onematrix_{M_{n}})\n $.\n Now, since $\\tilde{\\pi} \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n is a unital ${}^{\\ast}$-algebra representation,\n it is necessarily contractive.\n It follows that $\\funcCalcCts\\otimes\\mathrm{\\textit{id}}_{M_{n}}$ is contractive for each $n\\in\\naturals$.\n Thus $\\normCb{\\funcCalcCts} \\leq 1$.\n\n \\paragraph{Sufficiency:}\n If $\\funcCalcCts$ is completely bounded with $\\normCb{\\funcCalcCts} \\leq 1$,\n then, since $\\funcCalcCts$ is a (contractive!) unital homomorphism\n defined on the unital subalgebra\n ${\n \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}\n \\subseteq \\mathcal{A}\n \\subseteq \\BoundedOps{\\HilbertRaum_{0}}\n }$,\n one may apply the dilation theorem in \\cite[Theorem~4.8]{Pisier2001bookCBmaps}%\n \\footnote{\n This result is based on the factorisation of completely bounded maps\n (see \\cite[Theorem~4.8]{Pisier2001bookCBmaps}),\n which builds on Stinespring's theorem\n and can be attributed to\n Wittstock \\cite{Wittstock1981ArticleDilationDe,Wittstock1984inCollDilationEn},\n Haagerup \\cite{Haagerup1985inCollCbMaps},\n as well as Paulsen.\n }\n and obtain\n a Hilbert space $\\HilbertRaum_{1}$,\n an isometry $r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}$,\n and\n a ${}^{\\ast}$-algebra representation\n ${\\pi \\colon \\BoundedOps{\\HilbertRaum_{0}} \\to \\BoundedOps{\\HilbertRaum_{1}}}$\n such that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n \\funcCalcCts(a) = r^{\\ast}\\,\\pi(a)\\,r\n \\end{eqnarray}\n\n \\noindent\n for all $a\\in\\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}$.\n Since $\\mathcal{A}$ is unital,\n we can replace $\\HilbertRaum_{1}$\n with the $\\pi$-invariant subspace $\\quer{\\pi(\\mathcal{A})r\\HilbertRaum}$,\n which contains $r\\HilbertRaum$.\n In particular, one can assume that $\\pi$ is a unital\n (and thus non-degenerate) ${}^{\\ast}$-representation\n of $\\mathcal{A}$ on $\\HilbertRaum_{1}$\n and that $\\pi(\\mathcal{A})r\\HilbertRaum$ is dense in $\\HilbertRaum_{1}$.\n Since $L^{1}(G)$ is $\\norm{\\cdot}_{\\ast}$-dense in $C^{\\ast}(G)$,\n it follows that\n $(\\complex\\cdot\\onematrix + \\pi(L^{1}(G)))r\\HilbertRaum$\n is dense in $\\HilbertRaum_{1}$.\n\n By the correspondence between unitary representations and non-degenerate ${}^{\\ast}$-representations\n in abstract harmonic analysis\n (see \\Cref{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}),\n there exists a (unique) \\topSOT-continuous unitary representation\n ${U:G \\to \\BoundedOps{\\HilbertRaum_{1}}}$\n such that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n U(x)\\pi(f) = \\pi(L_{x}f) = \\pi(f(x^{-1}\\cdot))\n \\end{eqnarray}\n\n \\noindent\n for all $x \\in G$ and all $f \\in L^{1}(G)$.\n Our goal is to show that $(\\HilbertRaum_{1},U,r)$ is a unitary dilation of $T$.\n\n Let $x \\in M$ and $f \\in \\LOneCPlus{M}{G}$ be arbitrary.\n Then\n $L_{x}f = f(x^{-1}\\cdot) \\in \\LOneCPlus{M}{G}$,\n since\n $\\quer{\\mathop{\\textup{supp}}}(f(x^{-1}\\cdot)) = x \\cdot \\quer{\\mathop{\\textup{supp}}}(f) \\subseteq M$.\n Applying the construction of $\\funcCalcCts$ yields\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n r^{\\ast}\\,U(x)\\,\\pi(f)\\,r\n &\\eqcrefoverset{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n &r^{\\ast}\\,\\pi(\\Fourier (f(x^{-1}\\cdot)))\\,r\\\\\n &\\eqcrefoverset{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n &\\funcCalcCts(f(x^{-1}\\cdot))\\\\\n &= &\\displaystyle\n \\sotInt_{y \\in \\quer{\\mathop{\\textup{supp}}}(f(x^{-1}\\cdot))}\n f(x^{-1}y) T(y)\n \\:\\dee y\\\\\n &= &\\displaystyle\n \\sotInt_{y \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(y) T(xy)\n \\:\\dee y\\\\\n &= &T(x)\n \\cdot\n \\displaystyle\n \\sotInt_{y \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(y) T(y)\n \\:\\dee y\\\\\n &= &T(x)\\funcCalcCts(f).\\\\\n \\end{eqnarray*}\n\n Consider now the net\n $(f_{i})_{i \\in I} \\subseteq \\LOneCPlus{M}{G}$\n constructed\n in \\Cref{prop:func-calc-cts:basic:shift-unit:sig:article-stochastic-raj-dahya},\n for which it holds that\n ${\\funcCalcCts(f_{i}) \\underset{i}{\\longrightarrow} \\onematrix}$\n \\wrt the \\topSOT-topology.\n By construction,\n $\\norm{f_{i}}_{\\ast} \\leq 1$\n for each $i\\in I$,\n and since $\\pi$ is a ${}^{\\ast}$-algebra representation,\n it follows that\n $\\norm{\\pi(f_{i})} \\leq \\norm{f_{i}}_{\\ast} \\leq 1$\n for all $i\\in I$.\n We now claim that\n ${\\pi(f_{i})r \\underset{i}{\\longrightarrow} r}$\n \\wrt the \\topWOT-topology.\n Since $(\\pi(f_{i}))_{i \\in I}$ is uniformly bounded\n and ${(\\complex\\cdot\\onematrix + \\pi(L^{1}(G)))r\\HilbertRaum}$\n is dense in $\\HilbertRaum_{1}$,\n it suffices to show that\n ${\n \\brkt{\\pi(f_{i})r\\xi}{ar\\eta}\n \\underset{i}{\\longrightarrow}\n \\brkt{r\\xi}{ar\\eta}\n }$\n for\n $a\\in\\complex\\cdot\\onematrix + \\pi(L^{1}(G))$\n and\n $\\xi,\\eta\\in\\HilbertRaum$.\n To this end, consider\n an arbitrary $\\eta\\in\\HilbertRaum$\n and\n $a = c\\cdot\\onematrix + \\pi(g)$\n for arbitrary $c\\in\\complex$, $g\\in L^{1}(G)$.\n By the properties of the construction in\n \\Cref{prop:func-calc-cts:basic:shift-unit:sig:article-stochastic-raj-dahya}\n one has\n ${\\funcCalcCts(f_{i}) \\underset{i}{\\longrightarrow} \\onematrix}$\n \\wrt the \\topSOT-topology\n as well as\n ${\\norm{g^{\\ast} \\ast f_{i} - g^{\\ast}}_{\\ast} \\underset{i}{\\longrightarrow} 0}$\n and thus\n ${\\pi(g^{\\ast} \\ast f_{i}) \\underset{i}{\\longrightarrow} \\pi(g^{\\ast}) = \\pi(g)^{\\ast}}$\n in norm.\n Hence\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\\pi(f_{i})r\\xi}{ar\\eta}\n &= &c^{\\ast}\\brkt{\\pi(f_{i})r\\xi}{r\\eta}\n + \\brkt{\\pi(f_{i})r\\xi}{\\pi(g)r\\eta}\\\\\n &= &c^{\\ast}\\brkt{r^{\\ast}\\pi(f_{i})r\\xi}{\\eta}\n + \\brkt{\\pi(g)^{\\ast}\\pi(f_{i})r\\xi}{r\\eta}\\\\\n &\\eqcrefoverset{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n &c^{\\ast}\\brkt{\\funcCalcCts(f_{i})\\xi}{\\eta}\n + \\brkt{\\pi(g^{\\ast} \\ast f_{i})r\\xi}{r\\eta}\\\\\n &\\underset{i}{\\longrightarrow}\n &c^{\\ast}\\brkt{\\onematrix\\,\\xi}{\\eta}\n + \\brkt{\\pi(g)^{\\ast}r\\xi}{r\\eta}\n = \\brkt{r\\xi}{ar\\eta},\\\\\n \\end{eqnarray*}\n\n \\noindent\n from which the claim follows.\n Taking weak limits in the above computation\n applied to the $f_{i}$ thus yields\n $\n r^{\\ast}\\,U(x)\\,r = T(x)\\cdot\\onematrix\n $\n for all $x \\in M$.\n Hence $(\\HilbertRaum_{1},U,r)$ is a unitary dilation of $T$.\n \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[The discrete functional calculus]{The discrete functional calculus}\n\\label{sec:functional-calculus:discrete:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor a (not necessarily locally compact!) topological group $G$, we consider\n\n \\vspace{-3\\parskip}\n $$\n c_{00}(G)\n =\n \\{\n f \\in \\ell^{1}(G)\n \\mid\n \\mathop{\\textup{supp}}(f)~\\text{finite}\n \\},\n $$\n\n\\noindent\nwhich is a ${}^{\\ast}$-subalgebra of the convolution algebra $(\\ell^{1}(G),\\ast)$\nand thus of the unital $C^{\\ast}$-algebra $\\quer{\\ell^{1}(G)}$.\nLet $M \\subseteq$ be an arbitrary submonoid\nand suppose that $(G,M,\\cdot^{+})$ is a positivity structure\n(see \\Cref{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}).\nFor a (not necessarily \\topSOT-continuous) homomorphism,\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$,\non $\\HilbertRaum$, consider the map\n ${\\funcCalcDiscr \\colon c_{00}(G) \\to \\BoundedOps{\\HilbertRaum}}$\ndefined by\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:functional-calc:discr:sig:article-stochastic-raj-dahya}\n \\funcCalcDiscr(f)\n = \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n f(x)T(x^{-})^{\\ast}T(x^{+})\n \\end{eqnarray}\n\n\\noindent\nfor $f\\in c_{00}(G)$.\nOne can readily check that $\\funcCalcDiscr$ is a\nlinear, self-adjoint, unital map\nwhich satisfies\n $\n \\funcCalcDiscr(f \\ast g)\n = \\funcCalcDiscr(f)\n \\funcCalcDiscr(g)\n $\nfor $f,g\\in c_{00}(G)$.\nWe shall refer to this unital ${}^{\\ast}$-algebraic homomorphism\nas the \\highlightTerm{discrete functional calculus}\nassociated with $T$.\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Characterisation of regular dilations \\`{a} la Sz.-Nagy]\n\\makelabel{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya}\n Let $(G,M,\\cdot^{+})$ be a positivity structure\n where $G$ is a topological group\n and $M \\subseteq G$ is a submonoid.%\n \\footref{ft:disc-func-calc:G-not-locally-compact:sig:article-stochastic-raj-dahya}\n Further let\n ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n be an \\topSOT-continuous homomorphism.\n Then $T$ has a regular unitary dilation if and only if $\\funcCalcDiscr$ is completely positive.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\n\\footnotetext[ft:disc-func-calc:G-not-locally-compact:sig:article-stochastic-raj-dahya]{\n Note that we neither require $G$ to be locally compact\n nor $M$ to be a measurable subset in this theorem!\n}\n\nParts of the proof of \\Cref{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya}\nare similar to \\cite[Theorem~7.1~b)]{Nagy1970}.\nHowever, there are two main differences.\nFirstly, Sz.-Nagy works with extensions of $T$ to all of $G$,\nwithout explicitly defining this\n(except in the special cases of $G=\\reals^{d}$ and $G=\\integers^{d}$).\nSecondly, our approach relies on Stinespring's dilation theorem for $C^{\\ast}$-algebras,\nwhilst Sz.-Nagy's approach is more directly connected the theory of unitary representations.\n\n \\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya}\n \\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n Let $\\mathcal{A} \\colonequals \\quer{\\ell^{1}(G)}$\n be the (unital) group $C^{\\ast}$-algebra\n for the discretised version of $G$.\n Note that $c_{00}(G)$ is unital and self-adjoint,\n and thus constitutes an \\emph{operator system}\n (\\cf \\cite[Chapter~2, p.~9]{Paulsen1986bookCBmapsAndDilations}).\n\n \\paragraph{Necessity:}\n Suppose that $(\\HilbertRaum_{1},U,r)$ is a regular unitary dilation of $T$.\n By the correspondence between unitary representations and non-degenerate ${}^{\\ast}$-representations\n in abstract harmonic analysis\n (see \\Cref{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}),\n there exists a non-degenerate ${}^{\\ast}$-representation\n ${\n \\pi \\colon\n \\mathcal{A}\n \\to\n \\BoundedOps{\\HilbertRaum_{1}}\n }$\n such that \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya} holds.\n Using the regular unitary dilation and the ${}^{\\ast}$-representation,\n one obtains\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\\funcCalcDiscr(f)\\xi}{\\eta}\n &= &\\brktLong{\n \\Big(\n \\displaystyle\n \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n f(x)\n \\underbrace{\n T(x^{-})^{\\ast}T(x^{+})\n }_{=r^{\\ast}U(x)r}\n \\Big)\n \\xi\n }{\\eta}\\\\\n &= &\\displaystyle\n \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n f(x)\n \\brkt{U(x)r\\xi}{r\\eta}\\\\\n &\\eqcrefoverset{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}{=}\n &\\brkt{\\pi(f)r\\xi}{r\\eta}\\\\\n &= &\\brkt{r^{\\ast}\\,\\pi(f)\\,r\\xi}{\\eta}\n \\end{eqnarray*}\n\n \\noindent\n for all\n $f \\in c_{00}(G)$,\n $\\xi,\\eta\\in\\HilbertRaum$.\n Thus $\\funcCalcDiscr(a) = r^{\\ast}\\,\\pi(a)\\,r$\n for all $a\\in c_{00}(G)$.\n For $n\\in\\naturals$ and positive matrices\n $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(c_{00}(G))$\n it follows that\n $\n (\\funcCalcDiscr \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n = (\\funcCalcDiscr(a_{ij}))_{ij}\n = \\Big(\n r^{\\ast}\\,\\pi(a_{ij})\\,r\n \\Big)_{ij}\n = (r \\otimes \\onematrix_{M_{n}})^{\\ast}\n (\\pi \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n (r \\otimes \\onematrix_{M_{n}})\n $,\n which is positive,\n since\n ${\\pi \\otimes \\mathrm{\\textit{id}}_{M_{n}}}$\n is a ${}^{\\ast}$-representation of\n the $C^{\\ast}$-algebra $M_{n}(\\mathcal{A})$\n and thus positive.\n Thus $\\funcCalcDiscr$ is completely positive.\n\n \\paragraph{Sufficiency:}\n Since $c_{00}(G) \\subseteq \\mathcal{A}$ is an operator system,\n Averson's extension theorem\n (see \\cite[Theorem~7.5]{Paulsen1986bookCBmapsAndDilations})\n yields an extension of $\\funcCalcDiscr$ to a completely positive map\n between $\\mathcal{A}$ and $\\BoundedOps{\\HilbertRaum}$.\n Stinespring's dilation theorem\n (see \\cite[Theorem~1 and \\S{}3.~Remarks]{Stinespring1955dilation})\n applied to this yields\n a Hilbert space $\\HilbertRaum_{1}$,\n an isometry $r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}$,\n and\n a unital ${}^{\\ast}$-representation ${\\pi \\colon \\mathcal{A}\\to\\BoundedOps{\\HilbertRaum_{1}}}$,\n such that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n \\funcCalcDiscr(a) = r^{\\ast}\\,\\pi(a)\\,r\n \\end{eqnarray}\n\n \\noindent\n holds for all $a\\in c_{00}(G)$.\n Since $c_{00}(G)$ is a dense, unital ${}^{\\ast}$-subalgebra of $\\mathcal{A}$,\n we can replace $\\HilbertRaum_{1}$ by the $\\pi$-invariant closed subspace,\n $\\quer{c_{00}(G)r\\HilbertRaum}$, which contains $r\\HilbertRaum$.\n\n Since for all $x,y \\in G$\n one has\n $\\delta_{x}\\ast\\delta_{y} = \\delta_{xy}$\n and\n $\\delta_{x}\\in c_{00}(G)\\subseteq\\mathcal{A}$\n are unitary,%\n \\footnote{\n since\n $\\delta_{x}^{\\ast}\\ast\\delta_{x} = \\delta_{x^{-1}x} = \\delta_{e}$\n and\n $\\delta_{x}\\ast\\delta_{x}^{\\ast} = \\delta_{xx^{-1}} = \\delta_{e}$.\n }\n and since $\\pi$ is a unital ${}^{\\ast}$-representation,\n it follows that ${U \\colon G \\to \\BoundedOps{\\HilbertRaum}}$\n defined by $U(x) \\colonequals \\pi(\\delta_{x})$\n is a unitary homomorphism of $G$ on $\\HilbertRaum$.\n Moreover by \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n T(x^{-})^{\\ast}T(x^{+})\n = \\funcCalcDiscr(\\delta_{x})\n = r^{\\ast}\\,\\pi(\\delta_{x})\\,r\n = r^{\\ast}\\,U(x)\\,r\n \\end{eqnarray}\n\n \\noindent\n for all $x \\in G$.\n To show that $(\\HilbertRaum_{1},U,r)$ is a regular unitary dilation of $T$,\n it thus remains to show that $U$ is $\\topSOT$-continuous.\n\n To this end, first note that\n ${\n (\\BoundedOps{\\HilbertRaum},\\topSOT)\n \\times (\\BoundedOps{\\HilbertRaum},\\topSOT)\n \\ni (R, S)\n \\mapsto\n S^{\\ast}R \\in (\\BoundedOps{\\HilbertRaum},\\topWOT)\n }$\n is continuous.%\n \\footnote{\n This follows from the estimate\n $\n \\abs{\\brkt{\n (S_{2}^{\\ast}R_{2}-S_{1}^{\\ast}R_{1})\\xi\n }{\\eta}}\n =\n \\abs{\\brkt{\n (\n S_{1}^{\\ast}\n (R_{2}-R_{1})\n +\n (S_{2}-S_{1})^{\\ast}\n R_{1}\n +\n (S_{2}-S_{1})^{\\ast}\n (R_{2}-R_{1})\n )\n \\xi\n }{\n \\eta\n }}\n \\leq\n \\norm{(R_{2}-R_{1})\\xi}\n \\norm{S_{1}\\eta}\n +\n \\norm{R_{1}\\xi}\n \\norm{(S_{2}-S_{1})\\eta}\n +\n \\norm{(R_{2} - R_{1})\\xi}\n \\norm{(S_{2}-S_{1})\\eta}\n $\n for $R_{1},R_{2},S_{1},S_{2}\\in\\BoundedOps{\\HilbertRaum}$\n and $\\xi,\\eta\\in\\HilbertRaum$.\n By the form of this estimate,\n we do not need to restrict the operators to bounded subsets\n (\\cf \\cite[Lemma~3.1]{Ptak1985}).\n }\n So since ${T \\colon M \\to (\\BoundedOps{\\HilbertRaum},\\topSOT)}$\n and\n $\\cdot^{+}$ (and thus $\\cdot^{-}$)\n are continuous,\n it follows that\n ${\n G \\ni x\n \\mapsto\n r^{\\ast}U(x)r\n \\eqcrefoverset{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n T(x^{-})^{\\ast}T(x^{+})\n \\in \\BoundedOps{\\HilbertRaum}\n }$\n is \\topWOT-continuous.\n This implies that\n\n \\vspace{-3\\parskip}\n $$\n G \\ni x\n \\mapsto\n \\begin{array}[t]{0l}\n \\brkt{U(x)\\pi(\\delta_{y})r\\xi}{\\pi(\\delta_{z})r\\eta}\\\\\n = \\brkt{U(x)U(y)r\\xi}{U(z)r\\eta}\\\\\n = \\brkt{r^{\\ast}U(z^{-1}xy)r\\xi}{r\\eta}\\\\\n \\end{array}\n $$\n\n \\noindent\n is continuous\n for all $\\xi,\\eta\\in\\BoundedOps{\\HilbertRaum}$, $y,z\\in G$,\n which in turn entails the continuity of\n ${\n G \\ni x \\mapsto \\brkt{U(x)\\pi(f)r\\xi}{\\pi(g)r\\eta}\n }$\n for all $\\xi,\\eta\\in\\BoundedOps{\\HilbertRaum}$, $f,g \\in c_{00}(G)$.\n Since $U$ is unitary-valued and $\\pi(c_{00}(G))r\\HilbertRaum$ is dense in $\\HilbertRaum_{1}$,\n it follows that $U$ is a \\topWOT- and thus indeed an \\topSOT-continuous\n unitary representation of $G$ on $\\HilbertRaum_{1}$.\n \\end{proof}\n\n\\begin{rem}\n The \\topSOT-continuity of $T$ was only used\n to prove the \\topSOT-continuity of the unitary representation.\n Without this assumption, the above proof shows that\n $T$ has a (not necessarily \\topSOT-continuous) regular unitary dilation\n if and only if $\\funcCalcDiscr$ is completely positive.\n\\end{rem}\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Second main result]{Second main results: Unitary approximants}\n\\label{sec:second-results:sig:article-stochastic-raj-dahya}\n\n\\noindent\nThe functional calculi presented in \\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya}\nto characterise unitary and regular unitary dilations respectively,\nprovide us the means to study topological approximations of classical dynamical systems.\nWe exploit these results to prove\n\\Cref{%\n thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n}.\n\n\n\\noindent\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}\n\\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n The implications \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3} are clear.\n\n \\paragraph{\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}:}\n Let\n $(U^{(\\alpha)})_{\\alpha\\in\\Lambda}$,\n be a net of \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$.\n Suppose that $(U^{(\\alpha)}\\restr{M})_{\\alpha\\in\\Lambda}$\n approximates $T$ in the \\emph{uniform weak} sense.\n We make use of the \\emph{continuous functional calculi}\n ${\n \\funcCalcCts,\\funcCalcCts^{(\\alpha)}\n \\colon\n \\LOneCPlus{M}{G} \\to \\BoundedOps{\\HilbertRaum}\n }$\n associated with $T$\n and each $U^{(\\alpha)}\\restr{M}$ respectively\n (see \\S{}\\ref{sec:functional-calculus:continuous:sig:article-stochastic-raj-dahya}).\n By the characterisation in\n \\Cref{lemm:cts-functional-calculus:sig:article-stochastic-raj-dahya},\n $\\normCb{\\funcCalcCts^{(\\alpha)}} \\leq 1$\n for each $\\alpha\\in\\Lambda$,\n and, in order to show that $T$ has a unitary dilation,\n it suffices to show that\n $\\funcCalcCts$\n is completely bounded\n with $\\normCb{\\funcCalcCts} \\leq 1$.\n To this end, first observe that for each\n $\n a = c\\cdot\\delta_{e} + f\n \\in \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}\n \\equalscolon \\mathcal{A}\n $,\n uniform weak convergence yields\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\n \\funcCalcCts^{(\\alpha)}(a)\n \\xi\n }{\\eta}\n &= &\\brktLong{\n \\Big(\n c\\onematrix\n +\n \\displaystyle\n \\sotInt_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)\n U^{(\\alpha)}(x)\n \\:\\dee x\n \\Big)\n \\xi\n }{\\eta}\\\\\n &= &c\\brkt{\\xi}{\\eta}\n +\n \\displaystyle\n \\int_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)\n \\brkt{U^{(\\alpha)}(x)\\xi}{\\eta}\n \\:\\dee x\\\\\n &\\underset{\\alpha}{\\longrightarrow}\n &c\\brkt{\\xi}{\\eta}\n +\n \\displaystyle\n \\int_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n f(x)\n \\brkt{T(x)\\xi}{\\eta}\n \\:\\dee x\\\\\n &= &\\brkt{\n \\funcCalcCts(a)\n \\xi\n }{\\eta}\\\\\n \\end{eqnarray*}\n\n \\noindent\n for all $\\xi,\\eta\\in\\HilbertRaum$.\n Thus\n ${\n \\funcCalcCts^{(\\alpha)}(a)\n \\underset{\\alpha}{\\longrightarrow}\n \\funcCalcCts(a)\n }$\n \\wrt the $\\topWOT$-topology\n for each $a\\in \\mathcal{A}$.\n It follows that\n ${\n (\\funcCalcCts^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n \\underset{\\alpha}{\\longrightarrow}\n (\\funcCalcCts \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n }$\n \\wrt the $\\topWOT$-topology\n for\n $n\\in\\naturals$\n and matrices\n $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(\\mathcal{A})$.\n Since\n $\\funcCalcCts^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n is a contraction for each $\\alpha\\in\\Lambda$ and each $n\\in\\naturals$,\n it follows that\n $\\funcCalcCts \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n is a contraction for each $n\\in\\naturals$.\n Thus $\\funcCalcCts$ is completely bounded\n with $\\normCb{\\funcCalcCts} \\leq 1$.\n\n \\paragraph{\\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}, under cardinality assumption:}\n By assumption, $G$ contains a dense subspace $D \\subseteq G$\n and $\\mathop{\\textup{dim}}(\\HilbertRaum) \\geq \\max\\{\\aleph_{0},\\card{D}\\}$.\n Without loss of generality, one can replace $D$ by a dense subgroup of $G$.\n Let\n $(\\HilbertRaum_{1},U,r)$\n be a regular unitary dilation of $T$.\n Let $B \\subseteq \\HilbertRaum$ be an orthonormal basis (ONB) for $\\HilbertRaum$\n and\n $\\kappa \\colonequals \\card{B} = \\mathop{\\textup{dim}}(\\HilbertRaum) \\geq \\max\\{\\aleph_{0},\\card{D}\\}$\n and consider\n\n \\vspace{-3\\parskip}\n $$\n \\HilbertRaum_{0} \\colonequals \\quer{\\mathop{\\textup{lin}}}\\{\n U(x)r\\xi\n \\mid\n x \\in D,\n \\xi \\in B\n \\},\n $$\n\n \\noindent\n which is a $U$-invariant subspace.\n By the above cardinality assumptions\n and elementary computations with infinite cardinals\n (see \\exempli \\cite[\\S{}I.10]{Kunen2011BookSetTh}),\n one has cardinality\n $\\card{\\HilbertRaum_{0}} = \\card{B} = \\kappa$.\n It follows that\n $(\\HilbertRaum_{0},U\\restr{\\HilbertRaum_{0}},r)$\n is a regular unitary dilation of $T$.\n Furthermore, since $r$ is isometric, we have\n $\n \\kappa\n = \\mathop{\\textup{dim}}(\\HilbertRaum)\n = \\mathop{\\textup{dim}}(\\mathop{\\textup{ran}}(r))\n \\leq \\mathop{\\textup{dim}}(\\HilbertRaum_{0})\n = \\kappa\n $.\n Thus $\\mathop{\\textup{dim}}(\\HilbertRaum_{0}) = \\kappa = \\mathop{\\textup{dim}}(\\HilbertRaum)$.\n So without loss of generality, one may assume that $\\HilbertRaum_{0} = \\HilbertRaum$.\n It follows that there exists\n an isometry ${r\\in\\BoundedOps{\\HilbertRaum}}$\n and\n an $\\topSOT$-continuous\n unitary representation $U$ of $G$ on $\\HilbertRaum$\n such that\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n T(x)\n =\n r^{\\ast}\\,U(x)\\,r\n \\end{eqnarray}\n\n \\noindent\n for all $x \\in M$.\n\n Now let\n $P \\subseteq \\BoundedOps{\\HilbertRaum}$\n be the index set consisting of finite projections on $\\HilbertRaum$,\n directly ordered by $p \\succeq q$ $:\\Leftrightarrow$ $\\mathop{\\textup{ran}}(p) \\supseteq \\mathop{\\textup{ran}}(q)$.\n Let $p \\in P$ be arbitrary\n and let $F_{p} \\subseteq \\HilbertRaum$ be a finite ONB for $\\mathop{\\textup{ran}}(p)$.\n Since $r$ is an isometry,\n $\\tilde{F}_{p} \\colonequals \\{r\\,e \\mid e\\in F_{p}\\}$\n is also a finite orthonormal family of vectors.\n Let\n $B_{p},\\tilde{B}_{p}^{\\prime} \\subseteq \\HilbertRaum$\n be ONBs extending $F_{p},\\tilde{F}_{p}$ respectively.\n Since $\\HilbertRaum$ is infinite dimensional\n and $F_{p},\\tilde{F}_{p}$ are finite,\n one has\n $\n \\card{B_{p} \\setminus F_{p}}\n = \\mathop{\\textup{dim}}(\\HilbertRaum)\n = \\card{\\tilde{B}_{p} \\setminus \\tilde{F}_{p}}\n $.\n Thus there exists a bijection\n ${f \\colon B_{p} \\setminus F_{p} \\to \\tilde{B}_{p} \\setminus \\tilde{F}_{p}}$.\n Thus\n $g \\colonequals r\\restr{F_{p}} \\cup f$\n is a bijection between $B_{p}$ and $\\tilde{B}_{p}$.\n This extends uniquely to a unitary operator\n $w_{p} \\in \\BoundedOps{\\HilbertRaum}$.\n By construction,\n ${w_{p}\\restr{F_{p}} = r\\restr{F_{p}}}$\n and thus by linearity\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:unitary-squash:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n w_{p}p = rp\n \\end{eqnarray}\n\n \\noindent\n for each $p \\in P$.\n Finally, set\n\n \\vspace{-3\\parskip}\n $$\n U^{(p)} \\colonequals w_{p}^{\\ast}\\,U(\\cdot)\\,w_{p}\n $$\n\n \\noindent\n for each $p \\in P$,\n which are clearly \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$.\n We demonstrate that the net,\n $(U^{(p)}\\restr{M})_{p \\in P}$,\n of \\topSOT-continuous homomorphisms,\n is an \\emph{exact weak approximation} of $T$.\n To this end, let\n $\\xi,\\eta\\in\\HilbertRaum$ be arbitrary.\n Let $p_{0}\\in\\BoundedOps{\\HilbertRaum}$\n be the projection onto $\\mathop{\\textup{lin}}\\{\\xi,\\eta\\}$.\n For each $p \\in P$ with $p \\succeq p_{0}$ one has\n that\n $p\\xi = \\xi$\n and\n $p\\eta = \\eta$.\n By \\eqcref{eq:unitary-squash:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya},\n $\n w_{p}^{\\ast}r \\xi\n = w_{p}^{\\ast} r p \\xi\n = w_{p}^{\\ast} w_{p} p \\xi\n = p \\xi\n = \\xi\n $\n and similarly\n $\n w_{p}^{\\ast}r \\eta\n = \\eta\n $.\n The dilation yields\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\n T(x)\n \\xi\n }{\\eta}\n &\\eqcrefoverset{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=} &\\brkt{\n r^{\\ast}\\,U(x)\\,r\n \\xi\n }{\\eta}\\\\\n &= &\\brkt{\n r^{\\ast}\n w_{p}\n U^{(p)}(x)\n w_{p}^{\\ast}\n r\n \\xi\n }{\\eta}\\\\\n &= &\\brkt{\n U^{(p)}(x)\n w_{p}^{\\ast}\n r\n \\xi\n }{w_{p}^{\\ast}r\\eta}\\\\\n &= &\\brkt{\n U^{(p)}(x)\n \\xi\n }{\\eta}\\\\\n \\end{eqnarray*}\n\n \\noindent\n for all $x \\in M$\n and $p \\succeq p_{0}$.\n Hence\n $(U^{(p)}\\restr{M})_{p \\in P}$\n is an exact weak approximation of $T$.\n\\end{proof}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n\\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n The implications \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4} are clear.\n\n \\paragraph{\\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}:}\n Let\n $(U^{(\\alpha)})_{\\alpha\\in\\Lambda}$,\n be a net of \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$.\n Suppose that $(U^{(\\alpha)}\\restr{M})_{\\alpha\\in\\Lambda}$\n approximates $T$ in the \\emph{pointwise regular weak} sense.\n We now make use of the \\emph{discrete functional calculi}\n ${\n \\funcCalcDiscr,\\funcCalcDiscr^{(\\alpha)}\n \\colon\n c_{00}(G) \\to \\BoundedOps{\\HilbertRaum}\n }$\n associated with $T$\n and each $U^{(\\alpha)}\\restr{M}$ respectively\n (see \\S{}\\ref{sec:functional-calculus:discrete:sig:article-stochastic-raj-dahya}).\n By the characterisation in\n \\Cref{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya},\n each\n $\\funcCalcDiscr^{(\\alpha)}$\n is completely positive\n and, in order to show that $T$ has a regular unitary dilation,\n it suffices to show that\n $\\funcCalcDiscr$\n is completely positive.\n To this end, first observe for $f\\in c_{00}(G)$,\n that pointwise regular weak convergence yields\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\n \\funcCalcDiscr^{(\\alpha)}(f)\n \\xi\n }{\\eta}\n &= &\\displaystyle\n \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n f(x)\n \\brkt{\n U^{(\\alpha)}(x^{-})^{\\ast}U^{(\\alpha)}(x^{+})\n \\xi\n }{\\eta}\\\\\n &\\underset{\\alpha}{\\longrightarrow}\n &\\displaystyle\n \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n f(x)\n \\brkt{\n T(x^{-})^{\\ast}T(x^{+})\n \\xi\n }{\\eta}\\\\\n &= &\\brkt{\n \\funcCalcDiscr(f)\n \\xi\n }{\\eta}\\\\\n \\end{eqnarray*}\n\n \\noindent\n for all $\\xi,\\eta\\in\\HilbertRaum$.\n Thus\n ${\n \\funcCalcDiscr^{(\\alpha)}(f)\n \\underset{\\alpha}{\\longrightarrow}\n \\funcCalcDiscr(f)\n }$\n \\wrt the $\\topWOT$-topology\n for each $f\\in c_{00}(G)$.\n It follows that\n ${\n (\\funcCalcDiscr^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n \\underset{\\alpha}{\\longrightarrow}\n (\\funcCalcDiscr \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n }$\n \\wrt the $\\topWOT$-topology\n for\n $n\\in\\naturals$\n and matrices\n $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(c_{00}(G))$.\n Since\n $\\funcCalcDiscr^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n is positive for each $\\alpha$ and each $n\\in\\naturals$,\n it follows that\n $\\funcCalcDiscr \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n is positive for each $n\\in\\naturals$.\n Thus $\\funcCalcDiscr$ is completely positive.\n\n \\paragraph{\\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}, under cardinality assumption:}\n The proof is analogous to proof of\n \\eqcref{it:1:thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\eqcref{it:2:thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}\n of\n \\Cref{thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}.\n Relying on the cardinality assumptions, the same arguments as above\n yield a regular unitary dilation of $T$ of the form $(\\HilbertRaum,U,r)$,\n \\idest\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray}\n \\label{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n T(x^{-})^{\\ast}T(x^{+})\n = r^{\\ast}\\,U(x)\\,r\n \\end{eqnarray}\n\n \\noindent\n for all $x \\in G$.\n The nets of unitary operators,\n $(w_{p})_{p \\in P}$,\n and of \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$,\n $(U^{(p)} \\colonequals w_{p}^{\\ast}U(\\cdot)w_{p})_{p \\in P}$,\n are constructed as above.\n For $\\xi,\\eta\\in\\HilbertRaum$,\n letting $p_{0}$ be the projection onto $\\mathop{\\textup{lin}}\\{\\xi,\\eta\\}$,\n one has again\n $w_{p}^{\\ast}r\\xi=\\xi$\n and\n $w_{p}^{\\ast}r\\eta=\\eta$\n for $p \\succeq p_{0}$.\n The regular dilation yields\n\n \\vspace{-3\\parskip}\n \\begin{eqnarray*}\n \\brkt{\n T(x^{-})^{\\ast}\n T(x^{+})\n \\xi\n }{\\eta}\n &\\eqcrefoverset{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n &\\brkt{\n r^{\\ast}\\,U(x)\\,r\n \\xi\n }{\\eta}\\\\\n &= &\\brkt{\n r^{\\ast}\n w_{p}\n U^{(p)}(x)\n w_{p}^{\\ast}\n r\n \\xi\n }{\\eta}\\\\\n &= &\\brkt{\n U^{(p)}(x)\n w_{p}^{\\ast}\n r\n \\xi\n }{w_{p}^{\\ast}r\\eta}\\\\\n &= &\\brkt{\n U^{(p)}(x)\n \\xi\n }{\\eta}\\\\\n \\end{eqnarray*}\n\n \\noindent\n for all $x \\in G$\n and $p \\succeq p_{0}$.\n Hence\n $(U^{(p)}\\restr{M})_{p \\in P}$\n is an \\emph{exact regular weak approximation} of $T$.\n\\end{proof}\n\n\n\nAs an immediate application of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya},\nwe demonstrate an infinite class of commuting systems\nwhich admit no regular weak unitary approximations.\nThe following examples demonstrate in particular,\nthat the problem of unitary approximability (in the \\emph{regular} case)\nof a commuting system cannot be reduced to the unitary approximability of strict subsystems.\n\n\\begin{cor}\n\\makelabel{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya}\n Let\n $d\\in\\naturals$ with $d \\geq 2$\n and\n $\\HilbertRaum$ be an infinite dimensional Hilbert space.\n Then there exists an infinite class\n of commuting families, $\\{T_{i}\\}_{i=1}^{d}$,\n of contractive $\\Cnought$-semigroups on $\\HilbertRaum$\n whose generators have strictly negative spectral bounds,%\n \\footnote{\n The \\highlightTerm{spectral bound} of a linear operator\n ${A \\colon \\mathop{\\textup{dom}}(A)\\subseteq\\HilbertRaum\\to\\HilbertRaum}$\n is given by\n $\\sup\\{\\mathop{\\mathfrak{R}\\mathrm{e}} \\lambda \\mid \\lambda\\in\\opSpectrum{A}\\}$\n (\\cf \\cite[Definition~1.12]{EngelNagel2000semigroupTextBook}).\n }\n such that\n $\\{T_{i}\\}_{i \\in C}$\n has an exact regular weak unitary approximation\n for each $C \\subsetneq \\{1,2,\\ldots,d\\}$,\n whilst\n $\\{T_{i}\\}_{i=1}^{d}$\n has no pointwise regular weak unitary approximation.\n\\end{cor}\n\nThe class of semigroups can be constructed as in\n\\cite[Proposition~5.3]{Dahya2023dilation}.\nFor the reader's convenience, we sketch the construction.\nWe apply the characterisation in\n\\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\nto $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$\n(see \\Cref{e.g.:monoids:positivity:sig:article-stochastic-raj-dahya})\nas well as the correspondence between\nbetween \\topSOT-continuous homomorphisms defined over $\\realsNonNeg^{d}$\nand commuting families of $\\Cnought$-semigroups\ndiscussed in \\S{}\\ref{sec:introduction:sig:article-stochastic-raj-dahya}.\n\n \\begin{proof}[of \\Cref{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya}]\n By assumption, one can find orthonormal closed subspaces\n $\\HilbertRaum_{1},\\HilbertRaum_{2}\\subseteq\\HilbertRaum$\n with\n $0 < \\mathop{\\textup{dim}}(\\HilbertRaum_{2}) \\leq \\mathop{\\textup{dim}}(\\HilbertRaum_{1})$\n such that\n $\\HilbertRaum=\\HilbertRaum_{1}\\bigoplus\\HilbertRaum_{2}$.\n Working \\wrt this partition,\n and letting\n $\\alpha \\in (\\frac{1}{\\sqrt{d}},\\:\\frac{1}{\\sqrt{d-1}})$\n be arbitrary,\n we consider bounded operators of the form\n\n \\vspace{-3\\parskip}\n $$\n A_{i} = -\\onematrix\n +\n \\begin{smatrix}\n \\zeromatrix &2\\alpha V_{i}\\\\\n \\zeromatrix &\\zeromatrix\\\\\n \\end{smatrix}\n $$\n\n \\noindent\n where\n $V_{i} \\in \\BoundedOps{\\HilbertRaum_{2}}{\\HilbertRaum_{1}}$\n can be chosen to be any isometry.\n One can show that $\\{A_{i}\\}_{i=1}^{d}$ is a commuting family of\n dissipative operators whose spectra are each given by $\\{-1\\}$.\n Thus, $\\{T_{i} \\colonequals (e^{t A_{i}})_{t\\in\\realsNonNeg}\\}_{i=1}^{d}$\n is a commuting family of contractive $\\Cnought$-semigroups,\n whose (bounded) generators have strictly negative spectral bounds.\n As in \\cite[Proposition~5.3]{Dahya2023dilation},\n it can be shown that\n $\\{A_{i}\\}_{i \\in C}$ is completely dissipative\n for each $C \\subsetneq \\{1,2,\\ldots,d\\}$,\n whilst\n $\\{A_{i}\\}_{i=1}^{d}$ is not completely dissipative.\n By the characterisation for semigroups with bounded generators\n (\\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}),\n it follows that\n $\\{T_{i}\\}_{i \\in C}$ has a simultaneous regular unitary dilation\n for each $C \\subsetneq \\{1,2,\\ldots,d\\}$,\n whilst\n $\\{T_{i}\\}_{i=1}^{d}$ does not.\n By \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya},\n the claim follows.\n \\end{proof}\n\n\\begin{rem}\n\\label{rem:conjecture-ordinary-unitary-dilation:sig:article-stochastic-raj-dahya}\n It is well known that commuting systems of $d=2$ contractive $\\Cnought$-semigroups\n have simultaneous unitary dilations\n (see\n \\cite{Slocinski1974},\n \\cite[Theorem~2]{Slocinski1982},\n and\n \\cite[Theorem~2.3]{Ptak1985}%\n ).\n Hence by\n \\Cref{%\n thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n }\n and \\Cref{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya},\n the topologies defined by\n exact weak\n (\\respectively uniform weak)\n convergence\n are in general%\n \\footnote{\n \\viz for commuting families of $d \\geq 2$ semigroups\n over infinite dimensional Hilbert spaces.\n }\n strictly weaker than their \\emph{regular} counterparts.\n In particular,\n the characterisations in\n \\Cref{%\n thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n }\n now provide a sharp topological distinction\n between \\emph{unitary dilations}\n and \\emph{regular unitary dilations}.\n\\end{rem}\n\nAt the start of this paper we mentioned two different ways to treat irreversible systems:\nvia embeddings into or approximations by reversible systems.\nThe characterisations in\n \\Cref{%\n thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n },\ndemonstrate that, unter moder conditions,\nthese are in fact equivalent\nfor respective choices of dilations and approximations.\nThis holds for the commutative setting\nand for a large class of classical dynamical systems,\nincluding dynamical systems consisting of\nfamilies of $\\Cnought$-semigroups on Hilbert spaces\nsatisfying a semigroup version of the CCR in Weyl-form\n(%\n see\n \\Cref{e.g.:non-commuting-family-heisenberg-d-C:sig:article-stochastic-raj-dahya},\n \\Cref{e.g.:monoids:e-joint:sig:article-stochastic-raj-dahya},\n and\n \\Cref{e.g.:monoids:positivity:sig:article-stochastic-raj-dahya}%\n).\n\n\n\n\n\\setcounternach{section}{1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\lb{intro}\nIn its weak-field and slow-motion approximation, general relativity predicts that, in addition to the time-honored post-Newtonian (pN) gravitoelectric and gravitomagnetic precessions induced by the mass $M$ (Schwarzschild) and the spin angular momentum $\\bds S$ (Lense-Thirring) of the central body acting as source of the gravitational field, other pN gravitoelectric and gravitomagnetic orbital effects related to its oblateness arise as well \\citep{Sof89,1988CeMec..42...81S,1990CeMDA..47..205H,1991ercm.book.....B,2014PhRvD..89d4043W,2014CQGra..31x5012P,2015IJMPD..2450067I,2015CeMDA.123....1M,2016JGeod..90.1345S,2018RSOS....580640F,2018CeMDA.130...40S}. So far, they have never been put to the test in any astronomical and astrophysical scenarios, despite some recent preliminary investigations pertaining the planet Jupiter in our solar system \\citep{2013CQGra..30s5011I,2019MNRAS.484.4811I}; for some embryonic thoughts about an Earth-spacecraft scenario, see \\citet{2013CQGra..30s5011I,2015IJMPD..2450067I}. To the pN level, the oblateness of astronomical bodies modifies also the propagation of the electromagnetic waves in their deformed spacetime. About the perspectives of measuring the resulting deflection due to Jupiter with astrometric techniques, see, e.g.,\n\\citet{2006CQGra..23.4853C},\\,\\citet{2007PhRvD..75f2002K},\\,\\citet{2008PhRvD..77d4029L},\\,\\citet{2019MNRAS.485.1147A}, and references therein.\n\nAn analysis of the analytical expressions of the pN gravitoelectric and gravitomagnetic orbital precessions due to the asphericity of the primary \\citep{2015IJMPD..2450067I,2019MNRAS.484.4811I} shows that the key ingredients needed to enhance their magnitude are a strongly distorted central body, and a highly eccentric and close orbit of the moving particle.\n\nBinaries composed by a pulsar and a main-sequence star \\citep{1998MNRAS.298...67W} may offer, in principle, interesting natural laboratories to try to investigate such little known pN effects.\nIndeed, they are systems composed of a neutron star regularly emitting electromagnetic radio pulses orbiting an usually more massive main sequence star, which, in most cases, is highly distorted due to its fast rotation, along a generally elliptical orbit.\nA very accurate observable quantity in binary pulsars is represented by the measurement of the times of arrival (TOAs) $\\tau$ of the pulses emitted by the neutron star which, in case it has a gravitationally bound companion, exhibit a regular variation $\\delta\\tau$ due to, among other things, the Keplerian motion about the common centre of mass: it is the R{\\o}mer-like time delay. The full variation of the pulses' times of arrival is due to several other effects connected, e.g., with the propagation of the electromagnetic waves through the deformed spacetime of the system \\citep{1998MNRAS.298...67W}.\n\nThe first binary pulsar hosting a main sequence star to be discovered was PSR B1259-63 \\citep{1992ApJ...387L..37J,2014MNRAS.437.3255S}; it is characterized by a highly eccentric orbit ($e=0.870$) with an orbital period of $P_{\\rm b}=1237~\\textrm{d}=3.38~\\textrm{yr}$. The pulsar's non-degenerate companion is the fast spinning Be star LS 2883, whose equatorial velocity $V_\\textrm{e}$ is about $280~\\textrm{km~s}^{-1}$ corresponding to $\\sim 70\\%$ of its break-up velocity \\citep{1996MNRAS.280L..31P}. Its mass $M$ and equatorial radius $R_\\textrm{e}$ amounts to about 30 Solar masses $(\\textrm{M}_\\odot)$ and $9.7$ Solar radii $(\\textrm{R}_\\odot)$ \\citep{negueruela+11}.\nLater, the eccentric ($e=0.808$) binary PSR J0045-7319 was discovered \\citep{1994ApJ...423L..43K}. To date, it is the fastest orbiting system since it is $P_{\\rm b}=51.17~\\textrm{d}$. Its primary is a main sequence B-star spinning close to its break-up velocity \\citep{1995ApJ...452..819L,1996Natur.381..584K}. PSR J1638-4725, having an orbital period of $P_{\\rm b}=1940~\\textrm{d}=5.3~\\textrm{yr}$ and $e=0.95$, was found by \\citet{2006MNRAS.372..777L}. Its stellar companion should be a rapidly rotating Be star. PSR J1740-3052 \\citep{2001MNRAS.325..979S}, with $P_{\\rm b}=231~\\textrm{d}$ and $e=0.578$, hosts most likely a B-type main-sequence star \\citep{2010MNRAS.406.1848T,2012MNRAS.425.2378M}. The most recently discovered main-sequence-star binary pulsar is the highly eccentric ($e=0.93$) PSR J2032+4127 \\citep{2015MNRAS.451..581L} characterized by $P_{\\rm b}=8578~\\textrm{d}=23.5~\\textrm{yr}$. The companion of the neutron star is the massive Be star MT91 213 with $M\\simeq 15\\,\\textrm{M}_\\odot$.\n\nFor the sake of completeness, we mention also a few other binary pulsars hosting a non-degenerate star, although they are not relevant for our purposes in view of the nature of their non massive and fast-rotating partners. They are PSR J1903+0327 \\textcolor{black}{\\citep{2018ApJS..235...37A}}, whose companion is a F5V-GOV $\\sim 1~\\textrm{M}_\\odot$ star moving in $P_{\\rm b}=95~\\textrm{d}$ along a rather eccentric orbit with $e=0.44$, the transitional millisecond pulsar PSR J1023+0038 \\citep{2009Sci...324.1411A} orbiting a low-mass ($0.2~\\textrm{M}_\\odot$) companion star in a circular path with $P_{\\rm b}=4.75~\\textrm{hr}$.\nThe rms timing residuals of the aforementioned binary pulsars are all of the order of $\\lesssim \\textrm{ms}$, apart from PSR J1903+0327\nwhich \\textcolor{black}{is} at the $\\simeq \\textcolor{black}{1}\\,\\upmu\\textrm{s}$ level; more specifically, they are $\\simeq 0.46~\\textrm{ms}$ over $13~\\textrm{yr}$ for PSR B1259-63 \\citep{2004MNRAS.351..599W}, $7.4~\\textrm{ms}$ over $2~\\textrm{yr}$ for PSR J0045-7319 \\citep{1994ApJ...423L..43K}, $\\simeq 5.3~\\textrm{ms}$ over $4.35~\\textrm{yr}$ for PSR J1638-4725 \\citep{2006MNRAS.372..777L}, $\\simeq 0.8~\\textrm{ms}$ over $2.29~\\textrm{yr}$ for PSR J1740-3052 \\citep{2001MNRAS.325..979S}, $\\simeq 0.5-1~\\textrm{ms}$ over about $6~\\textrm{yr}$ for PSR J2032+4127 \\citep{2015MNRAS.451..581L}, $\\simeq 1~\\textrm{\\upmu s}$ over about $3~\\textrm{yr}$ for PSR J1903+0327 \\textcolor{black}{\\citep{2018ApJS..235...37A}}, $0.1~\\textrm{ms}$ over about $4~\\textrm{yr}$ for PSR J1023+0038 \\citep{2013arXiv1311.5161A}.\nTable~\\ref{tavola1} summarizes the key data for the binary pulsars hosting massive, fast rotating main sequence stars.\n\\begin{sidewaystable}[ht]\n\\begin{center}\n\\small{\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n \\hline\nPulsar & Companion & Distance (kpc) & $M~(\\textrm{M}_\\odot)$ & $R_\\textrm{e}~(\\textrm{R}_\\odot)$ & $R_\\textrm{p}~(\\textrm{R}_\\odot)$ & $P_\\textrm{b}$ (d) & $e$ & $P$ (ms) & $\\upsigma_\\tau$ (ms) & $\\Delta T$ (yr) \\\\\n\\hline\nPSR J0045-7319 & B1V star & In SMC & $8.8\\pm 1.8$ & $6.4\\pm 0.7$ & $-$ & $51.17$ & $0.808$ & $930$ & $7.4$ & $2$ \\\\\nPSR J1740-3052 & Main sequence star & $-$ & $> 11$ & $-$ & $-$ & $231$ & $0.578$ & $570$ & $0.8$ & $2.29$ \\\\\nPSR B1259-63 & B2e star LS 2883 & $2.75$ & $\\simeq 30$ & $\\simeq 9.7$ & $\\simeq 8.1$ & $1237$ & $0.870$ & $48$ & $0.46$ & $13$ \\\\\nPSR J1638-4725 & Be star & $-$ & $>4$ & $-$ & $-$ & $1940$ & $0.95$ & $764$ & $5.3$ & $4.35$ \\\\\nPSR J2032+4127 & Be star MT91 213 & $1.7$ & $\\simeq 15$ & $-$ & $-$ & $8578$ & $0.93$ & $143$ & $0.5-1$ & $6$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{Binary pulsars with a massive, fast rotating main sequence star companion discovered so far. For each of them, we list the companion, the distance (when available), the mass $M$, the equatorial and polar radii $R_\\textrm{e},\\,R_\\textrm{p}$ (when available), the orbital period $P_\\textrm{b}$, the orbital eccentricity $e$, the spin period $P$, the rms timing residuals $\\upsigma_\\tau$, the data analysis time span $\\Delta T$. For the values of the listed parameters, see the references cited in the text and the online pulsar catalog at http:\/\/www.atnf.csiro.au\/research\/pulsar\/psrcat\/. For PSR B1256-63 we give the equatorial $R_\\textrm{e}$ and polar $R_\\textrm{p}$ radii as estimated by \\citet{negueruela+11}.}\\label{tavola1}\n\\end{sidewaystable}\n\nAbout the achievable accuracy level in timing residuals, in the case of the pulsars orbiting a main sequence star, their timing seems doomed to stay at the $\\simeq\\textrm{ms}$ level. It is so because, for evolutionary reasons, they are not fully recycled \\citep{2010NewAR..54...93S}. Thus, their spinning periods $P$ are not at the millisecond level, and their TOAs are not measured with a precision of the order of $\\simeq \\upmu\\textrm{s}$\\textcolor{black}{, and their timing is often contaminated by timing noise \\citep{Hobbsetal010}}. The timing of the non-recycled pulsars is almost always less accurate than for the fully recycled pulsars. Indeed, PSR J1903+0327 rotates with a period $P=2.5\\,\\textrm{ms}$, and its rms timing residuals are as little as $\\simeq 1\\,\\upmu\\textrm{s}$. The spin period of PSR J1023+0038 is $P=1.7\\,\\textrm{ms}$, and its rms timing residuals are $0.1\\,\\textrm{ms}$.\nIncidentally, we mention the fact that, according to Table\\,2 of\n\\citet{2018ApJS..235...37A}, the rms timing accuracy of some fully recycled pulsars with $P=1.5-10\\,\\textrm{ms}$, isolated or with a white dwarf as companion, is of the order of $0.1-0.2\\,\\upmu\\textrm{s}$. It is expected that future instrumental improvements may push the rms timing accuracy of some of them to the\\footnote{A. Possenti, personal communication to L.I., April 2019.} $\\simeq 10\\,\\textrm{ns}$ level over time spans some yr long. For binary pulsars hosting another neutron star, the rms timing accuracy is of the order of $1-100\\,\\upmu\\textrm{s}$.\n\nHere, we will preliminarily investigate the size and the temporal patterns of the perturbations $\\Delta\\ton{\\delta\\tau}$ induced on the R{\\o}mer-like orbital part of the pulsar's time delay $\\delta\\tau$ by both the standard (Schwarzschild and Lense-Thirring) and the oblateness-driven pN accelerations felt by a fictitious neutron star orbiting a highly distorted, fast rotating B-type main sequence star in view of a possible detection in new binaries that may eventually be discovered in the future.\nHowever, caution is in order before inferring too optimistic conclusions from a straightforward comparison of our simulated time series with the rms timing residuals listed in Table\\,\\ref{tavola1}. Even if the size of some pK signatures were to be larger than the $\\simeq\\textrm{ms}$ level, it does not necessarily mean that such effects will be measurable in the actual processing of the real observations. Indeed, careful, dedicated simulated data reductions and covariance analyses should be performed by explicitly modeling the signals of interest, estimating its characteristic parameters and inspecting the resulting correlations with the other parameters usually estimated. It should be kept in mind that, in principle, an unmodeled effect may be removed from the post-fit residuals, at least to a certain extent, being \\virg{absorbed} in the estimated values of the other parameters determined in the data reduction.\nThus, our investigation should be deemed just as a sensitivity analysis able to preliminarily explore the potential of the scenarios considered.\nWe will assume the validity of general relativity throughout the paper, which is organized as follows.\n\nIn Section~\\ref{J2S}, we discuss the magnitude of the angular momentum $S$ and the dimensionless quadrupole mass moment $J_2$ of typical fast rotating massive B-type stars. Section~\\ref{GR} is devoted to the numerical calculation of the perturbations $\\Delta\\ton{\\delta\\tau}$ induced on the pulsar's R{\\o}mer time delay by some post-Keplerian (pK) classical and pN accelerations. We do not calculate the propagation delays accounting, e.g., for the effects on the pulsar's travelling electromagnetic waves through the deformed spacetime of the B-type star. Section~\\ref{conclu} summarizes our findings and contains our conclusions. Once again, we stress the preliminary nature of our sensitivity investigation; we do not perform a full covariance analysis implying, e.g., the simulation of the pulsar's TOAs and their reduction along with parameter estimation.\n\\section{Quadrupole mass moment, flattening, and angular momentum of fast rotating main sequence B-type massive stars}\\lb{J2S}\n\nThe binary pulsars on which we shall focus presumably own a main sequence B-type\nmassive star. Such stars are mostly fast rotators \\cite[][]{levato+13} and\nare hence distorted by the centrifugal acceleration. Such a\ndistortion takes the mass distribution away from spherical symmetry\nand endows these stars with a quadrupolar and higher gravitational\nmoments. In the case of Be stars (namely B stars with emission lines),\nrotation is believed to be almost critical, namely\nthe rotational velocity is taken close ($> 70\\%$) to the Keplerian velocity at equator \\cite[e.g.][]{Rivinius2013_v21p69}. When\ncritical rotation is reached, the surface distortion is maximum and\nthe flattening $\\nu\\doteq (R_\\textrm{e}-R_\\textrm{p})\/R_\\textrm{e}$\\textcolor{black}{, expressed in terms of the equatorial and polar radii $R_\\mathrm{e},\\,R_\\mathrm{p}$, respectively,} is close to one third. For such\nstars the computation of the gravitational quadrupolar moment cannot\nbe done perturbatively as it is the case for the Sun, which is a slow\nrotator \\cite[e.g.][]{rozelot_etal01}. Modelling these stars requires\ntwo-dimensional models. Fortunately, self-consistent 2D-models have recently been\nachieved with the ESTER code \\cite[][]{ELR13,RELP16}. Compared to previous\n2D-models, ESTER models include self-consistently the baroclinicity of\nthe stellar envelopes and can thus predict the associated differential\nrotation. They therefore provide unambiguously the total angular momentum\nof the star given, for instance, its equatorial velocity.\n\nWith the ESTER code, we computed the parameters of three stellar models of 10, 15, and 30\\,$\\textrm{M}_\\odot$ as they can represent the companions of PSR J0045-7319, PSR J2032+4127 and PSR B1256-63 respectively. Since the evolutionary status of the stars is unknown but presumably on or close to the main sequence, we computed their steady state at ZAMS (Zero-Age Main Sequence) and at half-main sequence to get an idea of the effects of evolution. We use standard galactic metallicity $Z=0.02$ with the solar mixture (which may be approximate for PSR J0045-7319 which is in the SMC, known to be less metallic than the Galaxy). Results of ZAMS and evolved models are displayed in Tables~\\ref{Ester_zams}\\,to\\,\\ref{Ester_ev} respectively. There we give the total spin angular momentum $S$ and the dimensionless quadrupole mass moment $J_2$ along with other bulk parameters of the models. We recall that multipole gravitational moments $J_\\ell$ are defined by the multipole expansion of the gravitational potential of a mass $M$, namely\n\\begin{equation}\nU\\ton{\\bds r} = -\\frac{\\mu}{r}\\left[1 - \\sum_\\ell J_\\ell\\left(\\frac{R_\\textrm{e}}{r}\\right)^\\ell \\mathcal{P}_\\ell\\ton{\\xi}\\right],\\lb{potenz}\n\\end{equation}\nwhere $\\mu\\doteq GM$ is the star's gravitational parameter, $G$ is the Newtonian constant of gravitation, $\\xi\\doteq \\bds{\\hat{S}}\\bds\\cdot\\bds{\\hat{r}}$ is the cosine of the angle $\\theta$ between the directions of the body's spin axis and of an external point at $\\bds r$, $\\mathcal{P}_\\ell\\ton{\\xi}$ is the Legendre polynomial of degree $\\ell$.\nWe consider the mass distribution of the stellar models to be symmetric with respect to equator thus making the $J_\\ell$ of odd order all vanish. The remaining $J_{2p}$ can be computed with the integral expression\n\\begin{equation}\nJ_{2p} = -\\frac{1}{MR^{2p}_\\textrm{e}}\\int_{\\mathcal{V}} r^{2p}\\,\\mathcal{P}_{2p}\\ton{\\xi}\\,\\rho\\ton{\\bds r}\\,d^3\\bds r\n\\end{equation}\nwhere the integration is over the volume $\\mathcal{V}$ of the star. The same expression is given\nin, for instance, \\cite{helled+11}. Here we are especially interested in $J_2$ and $S$, namely in\n\\begin{equation}\nJ_2 = -\\frac{1}{MR^2_\\textrm{e}}\\int_{\\mathcal{V}} r^2\\,\\mathcal{P}_2\\ton{\\xi}\\,\\rho\\ton{\\bds r}\\,d^3\\bds r\\quad {\\rm and}\\quad S = \\int_{\\mathcal{V}}r^2(1-\\xi^2)\\,\\Xi\\ton{\\bds r}\\rho\\ton{\\bds r}\\,d^3\\bds r\n\\end{equation}\nwhich are directly computed from the ESTER models; $\\Xi\\ton{\\bds r}$ is the local angular speed.\n\nWe choose an angular rotation rate of 70\\% of the actual critical (Keplerian) angular velocity of the star. Such a rotation rate is typical of the nearby fast rotating stars that have been measured by interferometry. Their flattening is typically $\\sim0.2$ \\cite[e.g.][]{domiciano_etal14}, as our models.\n\\begin{table}[ht]\n\\caption{ESTER models for Zero-Age Main Sequence (ZAMS) stars with an equatorial angular velocity at\n70\\% of the critical angular velocity. $R_\\textrm{e}$ and $V_\\textrm{e}$ are the equatorial radius and\nvelocity, respectively, $S$ is the total spin angular momentum, $\\nu$ is the\nflattening, and $J_2$ is the dimensionless mass quadrupole moment. The metallicity is Z=0.02 and the hydrogen mass fraction is $X=0.7$. In our simulations, we use the parameters of the star with $15\\,\\textrm{M}_\\odot$ listed here.}\n\\label{Ester_zams}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n$M~(\\textrm{M}_\\odot)$ & $R_\\textrm{e}~(\\textrm{R}_\\odot)$ & $V_\\textrm{e}~(\\textrm{km}~\\textrm{s}^{-1})$ & $S~(\\times 10^{44}\\textrm{J}~\\textrm{s})$ & $\\nu$ & $J_2~(\\times 10^{-3})$\\\\\n\\hline\n 10 & 4.74 & 444 & 15.3 & 0.201 & 1.63 \\\\\n 15 & 5.96 & 485 & 34.1 & 0.203 & 1.92 \\\\\n 30 & 8.89 & 562 & 125. & 0.210 & 2.17 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{table}[ht]\n\\caption{Same as in table~\\ref{Ester_zams} but for ESTER models of\nstars at mid-main-sequence, namely when the hydrogen mass fraction in the convective core is half of the initial one.}\n\\label{Ester_ev}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n$M~(\\textrm{M}_\\odot)$ & $R_\\textrm{e}~(\\textrm{R}_\\odot)$ & $V_\\textrm{e}~(\\textrm{km}~\\textrm{s}^{-1})$ & $S~(\\times 10^{44}~\\textrm{J}~\\textrm{s})$ & $\\nu$ & $J_2~(\\times 10^{-3})$\\\\\n\\hline\n 10 & 6.93 & 367 & 13.3 & 0.203 & 0.816\\\\\n 15 & 9.07 & 393 & 28.0 & 0.207 & 0.788\\\\\n 30 & 14.8 & 435 & 870. & 0.227 & 0.495\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFrom Tables~\\ref{Ester_zams}~to~\\ref{Ester_ev}, we clearly see that as evolution proceeds, namely as the hydrogen content of the core decreases, $J_2$ decreases as expected from the resulting contraction of the convective core. From the work of \\cite{james64} we can compute $J_2$ for a polytrope of index $n=3$ with a similar flattening as the ESTER models. We find that $J_2\\simeq 2.1\\times10^{-3}$ which is quite similar to the ZAMS models. For evolved models one should use polytropes with a higher polytropic index, as they are more centrally condensed, and typically $n=3.43$ matches the ESTER evolved models as far as $J_2$ is concerned. This result may be useful for simulating the orbital evolution of binary pulsars since polytropic models are much easier to compute.\n\\section{The perturbations of the R{\\o}mer-type pulsar's time delay due to some pK Newtonian and pN accelerations}\\lb{GR}\nHere, we will assume a coordinate system centered in the binary's barycenter whose reference $z$-axis is directed along the line of sight from the binary to the observer, while the reference $\\grf{x,~y}$ plane spans the plane of the sky.\n\\subsection{The pulsar as a structureless, pointlike particle}\\lb{punto}\nWe will, first, consider the following pK accelerations experienced by a test particle moving with velocity $\\textcolor{black}{\\bds{v}}$ in the external field of an oblate body of mass $M$, equatorial and polar radii $R_\\textrm{e},~R_\\textrm{p}$, ellipticity $\\varepsilon \\doteq \\sqrt{1- R_\\textrm{p}^2\/R_\\textrm{e}^2}$, angular momentum $\\bds S$ and dimensionless quadrupole mass moment $J_2$. In Section\\,\\ref{esteso}, we will discuss the limits of validity of the point-particle approximation.\n\nTo the Newtonian level, the external potential of the distorted star at the position $\\bds r$ is, from \\rfr{potenz},\n\\begin{equation}\nU\\ton{\\bds r} = U_0 + \\Delta U_2 = -\\frac{\\mu}{r}\\qua{1-\\ton{\\frac{R_\\textrm{e}}{r}}^2 J_2 \\mathcal{P}_2\\ton{\\xi}},\n\\end{equation}\nwhere $\\mathcal{P}_2\\ton{\\xi} = \\ton{3\\xi^2 - 1}\/2$ is the Legendre polynomial of degree 2.\nThe classical acceleration due to $J_2$ is\n\\begin{equation}\n{\\bds A}^{\\textrm{N}J_2} = -\\bds\\nabla \\Delta U_{J_2}= \\frac{3\\mu J_2 R_\\textrm{e}^2}{2r^4}\\qua{\\ton{5\\xi^2 - 1}\\bds{\\hat{r}} - 2\\xi\\bds{\\hat{S}}}.\\lb{NJ2}\n\\end{equation}\n\nThe 1pN gravitoelectric Schwarzschild-like acceleration affecting the motion of a test particle in the static field of a nonrotating, spherically symmetric body is \\citep{2010ITN....36....1P}\n\\begin{equation}\n{\\bds A}^{\\textrm{1pN}M} = \\frac{\\mu}{c^2 r^2}\\qua{\\ton{\\frac{4\\mu}{r}-{\\mathrm{v}}^2}\\bds{\\hat{r}} + 4{\\mathrm{v}}_r\\textcolor{black}{\\bds{v}}},\\lb{1pNM}\n\\end{equation}\nwhere $c$ is the speed of light in vacuum, and ${\\mathrm{v}}_r \\doteq \\textcolor{black}{\\bds{v}}\\bds\\cdot\\bds{\\hat{r}}$ is the radial velocity of the test particle. \\Rfr{1pNM} is responsible for the formerly anomalous perihelion precession of Mercury whose explanation was the first empirical confirmation of general relativity \\citep{1915SPAW...47..831E}.\n\nThe 1pN gravitomagnetic Lense-Thirring acceleration in the stationary field due to the rotating primary is \\citep{2010ITN....36....1P}\n\\begin{equation}\n{\\bds A}^{\\textrm{1pN}S} \\lb{1pNS} = \\frac{2GS}{c^2 r^3}\\qua{3\\xi\\bds{\\hat{r}}\\bds\\times\\textcolor{black}{\\bds{v}} + \\textcolor{black}{\\bds{v}}\\bds\\times\\bds{\\hat{S}}}.\n\\end{equation}\nThe gravitomagnetic field of the Earth was unambiguously measured for the first time by the Gravity Probe B (GP-B) mission \\citep{2011PhRvL.106v1101E}. Tests of the Lense-Thirring orbital precessions with some terrestrial geodetic satellites are ongoing; see, e.g. \\citet{2013CEJPh..11..531R}, and \\citet{2015CQGra..32o5012L} and references therein for comprehensive reviews.\n\nThe 1pN gravitoelectric acceleration felt by a test particle in the field of an oblate body is \\citep{1988CeMec..42...81S, Sof89, 1991ercm.book.....B, 2014PhRvD..89d4043W,2015IJMPD..2450067I}\n\\begin{equation}\n{\\bds A}^{\\textrm{1pN}M J_2} \\lb{1pNMJ2} = \\frac{\\mu J_2 R_\\textrm{e}^2}{c^2 r^4}\\grf{\\frac{3}{2}\\qua{\\ton{5\\xi^2 - 1}\\bds{\\hat{r}} - 2\\xi\\bds{\\hat{S}} }\\ton{{\\mathrm{v}}^2 -\\frac{4\\mu}{r} }- 6\\qua{\\ton{5\\xi^2 - 1}{\\mathrm{v}}_r - 2\\xi{\\mathrm{v}}_S }\\textcolor{black}{\\bds{v}}\n- \\frac{2\\mu}{r}\\ton{3\\xi^2 - 1}\\bds{\\hat{r}}},\n\\end{equation}\nwhere ${\\mathrm{v}}_S \\doteq \\textcolor{black}{\\bds{v}}\\bds\\cdot\\bds{\\hat{S}}$ is the component of the particle's velocity along the direction of the primary's spin.\nNote that the parameter $J_2$ in \\rfr{1pNMJ2} is the same entering \\rfr{NJ2}, as per Equations 1 to 2 of \\citet{1988CeMec..42...81S}.\n\nThe 1pN gravitomagnetic acceleration imparted to a test particle by the spin octupole moment\\textcolor{black}{\\footnote{\\textcolor{black}{It will be shown that its effects are small enough to justify order-of-magnitude calculations, without need of detailed stellar models.}}} of a uniformly rotating homogenous oblate spheroid \\citep{2014CQGra..31x5012P,2015CeMDA.123....1M,2019MNRAS.484.4811I} can be cast into the compact form\n\\begin{equation}\n\\bds A^{\\textrm{1pN}SJ_2} = \\frac{3GSR_\\textrm{e}^2\\varepsilon^2 }{7c^2r^5}\\textcolor{black}{\\bds{v}}\\bds\\times\\grf{5\\xi\\qua{7\\xi^2 - 3}\\bds{\\hat{r}} + 3\\qua{1 - 5\\xi^2}\\bds{\\hat{S}}}.\\lb{1pNSJ2}\n\\end{equation}\n\nThe pK accelerations of \\rfrs{NJ2}{1pNSJ2} perturb the otherwise Keplerian motion of the binary causing a change $\\Delta\\ton{\\delta\\tau}$ of the regular variation $\\delta\\tau$ of the TOAs due to the relative orbital motion of the pulsar and the massive companion. It can be modeled as the\nratio of the projection of the barycentric orbit of the pulsar onto the line of sight to $c$ \\citep{1991PhRvL..66.2549D,2000ApJ...544..921K}. Thus, $\\Delta\\ton{\\delta\\tau}$ can be calculated by looking at the perturbations $\\Delta z$ induced by \\rfrs{NJ2}{1pNSJ2} on the $z$-component of the pulsar's barycentric orbital motion. We do that by numerically integrating the equations of motion of a fictitious pulsar \\textcolor{black}{with mass $M_\\mathrm{p}=1.4\\,\\mathrm{M}_\\odot$} having as a companion a Be-type star with $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~\\nu=0.203,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},\\,J_2 = 1.92\\times 10^{-3}$, as per Table~\\ref{Ester_zams}, for different values of its orbital configuration, determined by the initial values of the semimajor axis $a$, the eccentricity $e$, the orbital inclination $I$ to the plane of the sky, the longitude of the ascending node $\\Omega$, the argument of periastron $\\omega$, the true anomaly \\textcolor{black}{at epoch} $f_0$, and of the stellar spin axis characterized by its inclination $i$ to the line of sight, and the longitude $\\phi$ of the projection of the stellar spin onto the plane of the sky. For a chosen pK acceleration ${\\bds A}^\\textrm{pK}$, in order to compute its perturbation $\\Delta\\ton{\\delta\\tau}=\\Delta z\\ton{t}\/c$ over, say, 10 yrs, we perform two runs sharing the same initial conditions with and without ${\\bds A}^\\textrm{pK}$, calculate the resulting time series of $z\\ton{t}\/c$ and take their difference. Figures~\\ref{fig_NJ2}~to~\\ref{fig_1pNSJ2} depict our results for a given orbital configuration and the aforementioned Be-type main sequence star; in the panels of each Figure, we vary the Keplerian orbital elements and the orientation of $\\bds{\\hat{S}}$ in order to investigate the sensitivity to the parameter space of the adopted binary system. \\textcolor{black}{For the sake of a comparison, the Keplerian delay for the adopted reference orbital configuration lies in the range $-250\\,\\mathrm{s}\\lesssim\\delta\\tau\\lesssim 50\\,\\mathrm{s}$ over one orbital revolution.}\n\nFigure\\,\\ref{fig_NJ2} displays the Newtonian signatures due to the star's $J_2$. It can be noted that the resulting signal is strongly dependent on the initial value of the true anomaly, spanning the range from $50\\,\\textrm{s}$ to $-150\\,\\textrm{s}$. Instead, for the given value $f_0 = 228\\,\\textrm{deg}$, the sensitivity to the other orbital parameters is rather modest, amounting to about $10\\,\\textrm{s}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{NJ2}, in $\\textrm{s}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M=15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~J_2 = 1.92\\times 10^{-3},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_NJ2}\n\\end{center}\n\\end{figure}\n\nThe 1pN gravitoelectric Schwarzschild-like signatures due to the stellar mass monopole are reproduced in Figure\\,\\ref{fig_1pNM}.\nAlso in this case, the initial value of the true anomaly induces a marked variability in the decadal time series which ranges from to $-5\\,\\textrm{s}$ to $25\\,\\textrm{s}$. The other orbital parameters have less impact since the resulting variation of the signals is of the order of about $2\\,\\textrm{s}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_M_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_M_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_M_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_M_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_M_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_M_f.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNM}, in $\\textrm{s}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M=15~\\textrm{M}_\\odot$.}\\label{fig_1pNM}\n\\end{center}\n\\end{figure}\n\nThe 1pN gravitoelectric time series induced by the quadrupole mass moment $J_2$ of the star are the subject of Figure\\,\\ref{fig_1pNMJ2}.\nIn this case, the ranges of variation due to all the orbital parameters are rather similar, amounting to about $20-50\\,\\textrm{ms}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNMJ2}, in $\\textrm{ms}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~J_2 = 1.92\\times 10^{-3},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_1pNMJ2}\n\\end{center}\n\\end{figure}\n\nFigure\\,\\ref{fig_1pNS} shows the 1pN gravitomagnetic Lense-Thirring signatures due to the spin dipole moment of the star. They are mainly sensitive to the orbital inclination $I$ and to the spin's inclination $i$ to the line of sight which induces a variability as large as $10-12\\,\\textrm{ms}$. The ranges of variation induced by the other parameters are, instead, of the order of $1-5\\,\\textrm{ms}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNS}, in $\\textrm{ms}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M = 15~\\textrm{M}_\\odot,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_1pNS}\n\\end{center}\n\\end{figure}\n\nThe 1pN gravitomagnetic time series caused by the spin octupole moment of the star are depicted in Figure\\,\\ref{fig_1pNSJ2}.\nThe widest range of variability, $30-50\\,\\upmu\\textrm{s}$, is due to the inclination $I$, the node $\\Omega$, and the longitude $\\phi$ of the spin's projection onto the plane of the sky. We also checked that, for a pulsar orbiting in $\\simeq 20-30\\,\\textrm{d}$ a star as massive as those in the last lines of Tables\\,\\ref{Ester_zams}\\,to\\,\\ref{Ester_ev} with periastron distances of $r_\\textrm{min}\\simeq 1.2-1.1\\,R_\\textrm{eq}$, the magnitude of the timing signatures would reach the $\\simeq 1-10\\,\\textrm{ms}$ level.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNSJ2}, in $\\upmu\\textrm{s}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~\\nu=0.203,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_1pNSJ2}\n\\end{center}\n\\end{figure}\n\\subsection{The effects of the mass and spin multipoles of the pulsar}\\lb{esteso}\nUntil now, we have considered the neutron star as a structureless, point particle moving around its more massive companion.\nIn fact, a pulsar is an extended body with its own mass and spin multipole moments which, at least in principle, may have an impact on its orbital motion in a full two-body framework.\n\nThe modification of \\rfr{1pNM} for two finite bodies of masses $M_\\textrm{A},\\,M_\\textrm{B}$\nis \\citep{Sof89}\n\\begin{align}\n{\\bds A}^{\\textrm{1pN}M} = \\frac{\\mu_\\textrm{tot}}{c^2 r^2}\\grf{\\qua{\\ton{4+2\\zeta}\\frac{\\mu_\\textrm{tot}}{r}-\\ton{1+3\\zeta}{\\mathrm{v}}^2 +\\frac{3}{2}\\zeta{\\mathrm{v}}_r^2}\\bds{\\hat{r}} + \\ton{4-2\\zeta}{\\mathrm{v}}_r\\textcolor{black}{\\bds{v}}},\\lb{megapN}\n\\end{align}\nwhere $\\mu_\\textrm{tot}=GM_\\textrm{tot}=G\\ton{M_\\textrm{A}+M_\\textrm{B}}$, and $\\zeta\\doteq M_\\textrm{A}M_\\textrm{B}\/M_\\textrm{tot}^2$.\nBy taking the standard value $M_\\textrm{p} = 1.4\\,\\textrm{M}_\\odot$ for the mass of the pulsar, it is $\\zeta = 0.078$ for the Be-star assumed in Section\\,\\ref{punto}. It turns out that the introduction of $\\zeta$ in our numerical code changes the size of the time series of Figure\\,\\ref{fig_1pNM} by $\\simeq 0.1-0.2\\,\\textrm{s}$, while their temporal patterns remain essentially unchanged. Given the current level of accuracy in the timing residuals, such a discrepancy might be significative, and \\rfr{megapN} should be used instead of \\rfr{1pNM}.\n\nIn regard to the angular momentum, the spin $\\bds S$ of the Be-star in \\rfr{1pNS} should be replaced by the sum \\citep{1975PhRvD..12..329B}\n\\begin{equation}\n\\bds S\\rightarrow \\ton{1 + \\frac{3}{4}\\frac{M_\\textrm{p}}{M}}\\bds S +\\ton{1 + \\frac{3}{4}\\frac{M}{M_\\textrm{p}}}{\\bds S}_\\textrm{p},\\lb{pulsarS}\n\\end{equation}\nwhere ${\\bds S}_\\textrm{p}$ is the angular momentum of the pulsar.\nBy assuming, e.g., $S_\\textrm{p}= 3\\times 10^{40}\\,\\textrm{J}\\,\\textrm{s}$ as for PSR J0737-3039A \\citep{2003Natur.426..531B,2004Sci...303.1153L,Kramer2012,\nKehletal017,2018IAUS..337..128K}, it turns out that the magnitude of the second term in \\rfr{pulsarS} amounts to about $\\simeq 7\\times 10^{-5}$ of that of the first one. Moreover, it is $\\ton{3\/4}\\ton{M_\\textrm{p}\/M}=0.07$. Thus, as far as the 1pN gravitomagnetic Lense-Thirring effect is concerned, our scenario can be well approximated by a restricted two-body system with a spinning primary, and \\rfr{1pNS} is substantially adequate. Indeed, it turns out that rescaling the star's angular momentum in our numerical simulations as dictated by \\rfr{pulsarS} slightly modifies the size of the time series of Figure\\,\\ref{fig_1pNS} by just $\\simeq 1-1.5\\,\\textrm{ms}$.\n\nThe effect of the pulsar's quadrupole mass moment $J_2^\\textrm{p}$ can be accounted for by the replacement $M\\rightarrow M_\\textrm{tot}$ in \\rfr{NJ2} and by writing in its right-hand-side another term for $J_2^\\textrm{p}$ analogous to that for the stellar oblateness $J_2$. As a result, by introducing the dimensional quadrupole mass moment\n$Q_2 \\doteq -J_2 M R_\\textrm{e}^2$, the two terms in the right-hand-side of the resulting modified version of \\rfr{NJ2} are weigthed by \\citep{1975PhRvD..12..329B}\n\\begin{align}\n\\mathcal{Q}_2 \\lb{Qu2} & = \\ton{1+\\frac{M_\\textrm{p}}{M}}Q_2, \\\\ \\nonumber \\\\\n\\mathcal{Q}_2^\\textrm{p} & = \\ton{1+\\frac{M}{M_\\textrm{p}}}Q^\\textrm{p}_2.\n\\end{align}\nFor a neutron star, it is \\citep{1999ApJ...512..282L,2004MNRAS.350.1416B,2012PhRvL.108w1104P,2013ApJ...777...68B}\n\\begin{equation}\nQ_2^\\textrm{p} = -q\\frac{M_\\textrm{p}^3 G^2}{c^4},\n\\end{equation}\nwith $0.07\\lesssim q \\lesssim 3.507$ for a variety of Equations of State (EOSs).\nThus, we have\n\\begin{align}\nQ_2 & = -9.8\\times 10^{47}\\,\\textrm{kg}\\,\\textrm{m}^2, \\\\ \\nonumber \\\\\nQ_2^\\textrm{p} & = -q\\,1.2\\times 10^{37}\\,\\textrm{kg}\\,\\textrm{m}^2,\n\\end{align}\nso that $\\mathcal{Q}_2^\\textrm{p}\\simeq 10^{-10}\\,\\mathcal{Q}_2$. On the other hand, $M_\\textrm{p}\/M = 0.09$ in \\rfr{Qu2}.\nThus, for the Newtonian signature of the quadrupole mass moment, the restricted two-body scenario with an oblate primary is adequate in the present case, and the use of \\rfr{NJ2} is justified provided that the stellar quadrupole moment $Q_2$ is replaced by \\rfr{Qu2}. Indeed, it turns out that the introduction of $\\mathcal{Q}_2$ in the numerical integration changes the size of the time series in Figure\\,\\ref{fig_NJ2} by $\\simeq 1\\,\\textrm{s}$ which may not be neglected, given the current level in $\\upsigma_{\\tau}$.\n\nDespite \\rfr{1pNMJ2} and \\rfr{1pNSJ2} were derived so far only for the motion of a test particle around a spinning, oblate mass, there are no doubts that they are adequate to the scenario considered here.\n\nActually, general relativity predicts that, in general, there is also a self-force due to the spin angular momentum of an extended rotating body in motion in an external gravitational field which acts on it modifying its orbit through a spin-orbit coupling \\citep{1951RSPSA.209..248P,1974RSPTA.277..59D,1979GReGr..11..149B,2006PhRvD..74l4006M,Mathis010,2012GReGr..44..719I}.\nIn order to quickly evaluate the possible impact of such an effect in our case, let us proceed as follows. To the 1pN level, the precession $\\dot\\Omega_{S_\\textrm{p}}$ of, say, the node $\\Omega$, averaged over one orbital revolution of the spinning pulsar in its motion around the massive Be-type star, assumed nonrotating, is \\citep{2012GReGr..44..719I}\n\\begin{equation}\n\\dot\\Omega_{S_\\textrm{p}} = \\frac{3\\mu \\sigma_\\textrm{p}\\csc I\\ton{{\\bds{\\hat{\\sigma}}}_\\textrm{p}\\bds\\cdot\\bds{\\hat{m}}}}{2c^2a^3\\ton{1-e^2}^{3\/2}}.\\lb{spinor}\n\\end{equation}\nIn this expression, ${\\bds\\sigma}_\\textrm{p} = {\\bds S}_\\textrm{p}\/M_\\textrm{p}$ is the pulsar's spin angular momentum per unit mass, while $\\bds{\\hat{m}} =\\grf{-\\cos I\\sin\\Omega,\\,\\cos I\\cos\\Omega,\\,\\sin I}$ is a unit vector in the orbital plane perpendicular to the line of the nodes.\nLet us compare \\rfr{spinor} to the analogous precession induced by some of the effects previously considered in which the pulsar was treated as a point particle. The Lense-Thirring node rate, induced by \\rfr{1pNS} which is responsible for the $\\simeq\\textrm{ms}$ time series of Figure\\,\\ref{fig_1pNS},\nis \\citep{2012GReGr..44..719I}\n\\begin{equation}\n\\dot\\Omega_\\textrm{LT} = \\frac{2GS\\csc I\n\\ton{\\bds{\\hat{S}}\\bds\\cdot\\bds{\\hat{m}}}}{c^2 a^3\\ton{1-e^2}^{3\/2}}.\n\\end{equation}\nIt turns out that, in the case of a pulsar like, e.g., PSR J0737-3039A orbiting the Be-type star of the Figures\\,\\ref{fig_NJ2}\\,to\\,\\ref{fig_1pNSJ2}, it is $\\left|\\dot\\Omega_{S_\\textrm{p}}\/\\dot\\Omega_\\textrm{LT}\\right|\\simeq 10^{-5}$. Thus, we conclude that the spin-orbit self-force experienced by the pulsar is completely negligible in our scenario.\nIn fact, there is also a further self-force acting on an extended rotating body moving in an external gravitational field due to the spin-spin coupling between the angular momenta of the source and of the orbiter itself \\citep{1979GReGr..11..149B,2012GReGr..44..719I}. By following the same strategy for the spin-orbit coupling, the results in \\citet{2012GReGr..44..719I} allow to conclude that, in our case, such an effect is even smaller than the previous one, being of the order of $\\simeq 10^{-7}$ of, say, the $\\simeq \\textrm{ms}$-level Lense-Thirring signal.\n\n\\section{Summary and conclusions}\\lb{conclu}\nWe preliminarily explored the possibility of putting to the test several pK features of motion of Newtonian and pN origin in binaries hosting a pulsar and a massive, fast rotating, highly distorted main sequence star characterized by mass $M$, angular momentum $\\bds S$, equatorial and polar radii $R_\\textrm{e},~R_\\textrm{p}$, flattening $\\nu$, ellipticity $\\varepsilon$ and dimensionless quadrupole moment $J_2$. Indeed, in addition to the usual 1pN Schwarzschild and Lense-Thirring effects due to the mass monopole $\\ton{\\propto GM c^{-2}}$ and spin dipole $\\ton{\\propto GSc^{-2}}$ moments, respectively, of the distorted stellar field, there are also other 1pN orbital effects, induced by the mass quadrupole $\\ton{\\propto GMR^2_\\textrm{e}J_2 c^{-2}}$ and spin octupole $\\ton{\\propto GSR^2_\\textrm{e}\\varepsilon^2c^{-2}}$ moments, whose magnitudes may, perhaps, lie above the sensitivity threshold of the pulsar timing residuals in yet-to-be-discovered close binaries. However, the Newtonian perturbations due to $J_2$ are larger than the pN ones.\n\nIn order to perform a preliminary sensitivity analysis, we numerically investigated the orbital shifts $\\Delta\\ton{\\delta\\tau}$ induced over 10 yr by all of such pK perturbations on the otherwise Keplerian R{\\o}mer-type delay $\\delta\\tau$ in the pulsar's TOAs for a Be-type main sequence star characterized by $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~\\nu=0.203,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},\\,J_2 = 1.92\\times 10^{-3}$ orbited by a pulsar with an orbital geometry compatible $\\ton{P_{\\rm b}\\simeq 40-70~\\textrm{d}}$ with some of the tightest binaries of this kind out of those discovered so far. We also investigated the sensitivity of the pK timing shifts to the whole system's parameter space by varying both the orientation of the stellar spin axis and the orbital elements of the pulsar's orbit.\nIt turns out that the magnitude of the Newtonian signature due to $J_2$ can be as large as $\\lesssim 4-150~\\textrm{s}$,\nwhile the pN gravitoelectric quadrupolar one is $\\lesssim 10-40~\\textrm{ms}$.\nThe pN gravitoelectric (Schwarzschild-like) signal due to the stellar mass monopole can be as large as $\\lesssim 1.5-20~\\textrm{s}$ level.\nThe pN gravitomagnetic shifts due to the spin dipole (Lense-Thirring) and octupole moments, for the evaluation of whose size the knowledge of the stellar spin angular momentum $S$ and ellipticity $\\varepsilon$ is crucial, are of the order of $\\lesssim 0.5-6~\\textrm{ms}$, and $\\lesssim 5-20~\\upmu\\textrm{s}$, respectively.\nThe rms of the timing residuals of all the non-recycled, non-millisecond pulsars like those having a fast rotating main sequence star companion discovered so far are $\\lesssim \\textrm{ms}$ over $2-13~\\textrm{yr}$. It seems difficult that they can be substantially improved in the future. This implies that, in principle, all the pK effects considered fall within the potential measurability domain, except the pN spin octupole which is $\\simeq 1-2$ orders of magnitude weaker, at least for the orbital configurations and the star considered in this study. The pN Lense-Thirring signatures are just at the $\\simeq\\textrm{ms}$ level.\nThe different temporal patterns characteristic of the signals investigated may be helpful in separating them. It turns out that the tiniest pN effect may reach the $\\simeq 1-10\\,\\textrm{ms}$ level only for a very tight, eccentric binary $(P_\\textrm{b}\\simeq 20-30\\,\\textrm{d},\\,r_\\textrm{min}\\simeq 1.1-1.2\\,R_\\textrm{e})$ hosting a Be-star with $30\\,\\textrm{M}_\\odot,\\,S=125-870\\times 10^{44}\\,\\textrm{J\\,s}$.\n\nFinally, we stress the preliminary nature of our sensitivity analysis. To this aim, we remark that we did not compute the other kinds of time delay connected, e.g., with the propagation of the electromagnetic waves in the deformed spacetime. Moreover, we did not perform a full covariance analysis implying a simulation of the pulsar's TOAs, their reduction, and parameter estimation.\n\\section*{Acknowledgements}\nWe are grateful to A. Possenti for useful information. MR and ADS also acknowledge the strong support of the French Agence Nationale de la Recherche (ANR), under grant ESRR (ANR-16-CE31-0007-01).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{intro}\nSelf-assembly is a ubiquitous phenomenon in nature, producing complex structures in biology, chemistry and physics.\nExamples include DNA \\cite{winfree1998design,goodman2005rapid,rothemund2006folding,ke2012three}, protein quaternary structure \\cite{Levy:2006ez,villar2009self}, protein aggregation \\cite{Cohen:2013fd}, viruses \\cite{lavelle2009phase}, micelles \\cite{israelachvili1994self}, and thin films \\cite{krausch2002nanostructured}.\n\nTwo fundamental questions about a self-assembling system are: do the building blocks always form the same structure and does the assembly grow indefinitely.\nThe former property is referred to as the {\\em determinism} and the latter as {\\em boundedness} \\cite{Evo,ARXIV2015}.\nTogether they form the assembly classification.\n\nProtein complexes are a prominent example of deterministic, bound self-assembly.\nMisfolding or erroneous binding of mutated versions of such proteins can cause the self-assembly of a protein complex to become nondeterministic, unbound protein aggregation.\nThis in turn is the hallmark of a number of severe diseases, such as sickle-cell anemia and Alzheimer's \\cite{Cohen:2013fd,Bunn:1997de}.\n\nIn this paper we introduce a framework for establishing the determinism and boundedness of a given set of self-assembling building blocks.\nWe apply this approach to a previously studied lattice self-assembly model introduced in \\cite{Modularity} and studied further in \\cite{Evo,green,greengen,ARXIV2015,TES2016}.\nIn this model square tiles have attractive interfaces of different types, with interactions governed by a simple set of rules.\nThe final structures assembled are sets of connected lattice sites known as polyominoes.\n\nSuch tile self-assembly or polyomino models have been useful for the study of genotype-phenotype maps, where the specification of the building blocks can be viewed as a genotype and the resulting structure as the phenotype \\cite{Evo,green,greengen}.\nThis approach can be combined with genetic algorithms to model evolutionary processes \\cite{Evo}.\n\nDespite being an abstract model, this polyomino model can directly and meaningfully map to real biological self-assembly phenomena.\nFor example, the sickle-cell mutation of hemoglobin that leads to unbound protein aggregation can be modelled using polyominoes \\cite{green}.\nFurthermore, experimental implementations of the polyomino model have been realized using DNA tiles \\cite{TES2016}. \n\nThe assembly process previously used is fully stochastic, and may be sketched out as: (a) the structure is seeded with a randomly selected tile, (b) a random face on the structure is chosen, (c) a random tile is drawn with a random orientation, and (d) the drawn tile binds to the structure if the interfaces are interacting.\nSteps (b-d) are then repeated until no further attachments are possible and assembly terminates.\n\nIn this paper, we introduce a graph based approach to replace stochastic assembly as the primary method for identifying the determinism and boundedness of a given set of self-assembly building blocks.\nThe stochastic approach is inelegant and suffers from a compromise between accuracy and speed, whereas the graph theoretical approach offers a robust methodology that improves accuracy and speed.\n\nThis paper proceeds by discussing self-assembly basics in Section \\ref{sec:Lsa}. Assembly graphs and their construction from tile sets are introduced in Section \\ref{sec:AG}.\nClassification preserving transformations of assembly graphs which simplify graph features into deterministic motifs are discussed in Section \\ref{sec:ATAP}.\nAfter classifying the assembly graph, the structure is validated under steric constraints in Section \\ref{sec:CP}.\nExtensions to the method and general discussion are in Sections \\ref{sec:FE} and \\ref{sec:D} respectively.\n\nWith generalizations to other geometries and dimensions, there are potential applications in the study of protein complexes and protein aggregation, as well as bioengineering and nanotechnology.\n\n\\section{Lattice self-assembly}\\label{sec:Lsa}\nFollowing the model introduced in \\cite{Modularity}, a lattice self-assembly tile set consists of one or more tile types.\nEach side of a tile (the geometric face) has an interface type, with every tile type in the set exhibiting unique configurations of these interfaces.\nConventionally, interactions are defined with {\\bf 0} as non-interacting and {\\bf 1} $\\leftrightarrow$ {\\bf 2}, {\\bf 3} $\\leftrightarrow$ {\\bf 4}, etc. as interacting pairs.\nHowever, these interaction rules can take any arbitrarily complex form provided interactions are bidirectional.\nInteractions are infinite in strength, meaning two tiles bind irreversibly if the adjoined interfaces are interacting.\nThere is an infinite population of each tile type, precluding any stoichiometric limitations.\n\n\n\\subsection{Definitions of structure and determinism}\n\nA structure is defined as a set of connected tiles, each with an associated tile type, orientation, and unique lattice coordinates.\nStructure determinism can be defined in three distinct categories, listed in increasing order of strictness: shape, tile, and orientation determinism.\nStructures are defined independently of absolute position and rotation, and so lattice coordinate translations or rotations of entire structures are considered indistinguishable (structures are {\\em one-sided} polyominoes).\nAn example of a tile set and its assembled polymino is shown in Figure \\ref{determinism}.\n\n{\\bf Shape determinism} requires all produced structures to be the same one-sided polyomino.\n\n{\\bf Tile determinism} additionally requires corresponding coordinates between two structures to have matching tile types.\n\n{\\bf Orientation determinism} further requires matching relative inter-tile orientations.\n\nOrientation determinism is standard choice due to the impact that tile type and relative orientation could have on the function of the assembled structure. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{determinism}\n \\caption{A tile set (left) and its assembled polyomino (right).\n Each tile type in the set can be assigned a number, which is used to label the tile type in the polyomino.\n Rotating the label indicates the corresponding rotation of the tile during assembly.}\\label{determinism}\n\\end{figure}\n\n\\section{Assembly graphs}\\label{sec:AG}\nAssembly tile sets are represented using the notation $\\{\\text{tile type 1}, \\text{tile type 2}, \\ldots \\}$, where tile type ordering is irrelevant.\nTile types have cyclic symmetry and have a length fixed by the geometry, so square tiles have four faces and can be represented as $\\left( F_1, F_2,F_3,F_4 \\right)$, where $F_i$ indicates the interface type of face $i$.\nWe use the convention that faces are encoded clockwise starting from the top.\nAn example of this notation can be found in the caption of Figure \\ref{assemblygraphdemo}.\n\nThe {\\bf assembly graph} of a tile set is constructed with nodes for each face on every tile type.\nThe faces within each tile type form cliques with edges labeled as internal, while interactions between interfaces (both inter- and intra-tile) are encoded with edges labeled as external. \nHence an assembly graph can be represented using an edge-labeled pseudograph (multigraph with loops).\nA detailed assembly graph example is shown in Figure \\ref{assemblygraphdemo}.\nInternal edges only depend on geometry, and are not displayed in future examples for clarity.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.475\\textwidth]{graph_construction}\n \\caption{(a) The building blocks for the tile set $\\{\\left(1,3,0,0\\right),\\left(2,5,0,4\\right),\\left(0,0,0,6\\right)\\}$ explicitly labeled with interfaces and the assembled structure.\n (b) The corresponding complete assembly graph with black (dashed) internal edges, and gray (solid) external edges.}\\label{assemblygraphdemo}\n\\end{figure}\n\n\\subsection{Assembly graph terminology}\nAssembly graphs contain three features of interest: single interacting faces, branching points, and cycles.\nFigure \\ref{assemblygraph} shows examples of partial assembly graphs with such features.\n\n{\\bf Single interacting face} (SIF) tiles are tile type which have only a single face with one or more external edges.\n\n{\\bf Branching points} occur when a given face has multiple external edges.\nNon-SIF branching points occur if a tile type has a branching point and at least one other face with external edges.\nNon-SIF branching points frequently cause nondeterminism due to diverging assembly pathways.\n\n{\\bf Cycles} have two varieties with identical impact on assembly classification: inter- and intra-tile.\nInter-tile cycles are walks on the assembly graph which alternate stepping on external and internal edges.\nIntra-tile cycles are walks of only a single step on an external edge connecting to the same tile.\nCycles are the primary source of unboundedness, due to the potential to endlessly traverse (and thus assemble) around the cycle.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.475\\textwidth]{graph_features}\n \\caption{Five partial assembly graphs demonstrating various features.\n (a) SIF tile (b) SIF branching point (c) non-SIF branching point (d) two intra-tile cycles (e) inter-tile cycle.}\\label{assemblygraph}\n\\end{figure}\n\n\\subsection{Graph connectivity}\nIf an assembly graph contains disconnected components, it is trivially nondeterministic due to seed dependence.\nThis is due to the inability to assemble between disconnected components (by definition), and hence the structures formed by seeds in different components are nondeterministic.\n\nHowever, each connected component can be assessed independently for determinism.\nBiologically, this can be related to genotypes encoding multiple independent phenotypes.\n\n\\subsection{Fundamental Deterministic Graph}\\label{subsec:FDG}\n\nA {\\bf treelike} assembly graph is one without any cycles or non-SIF branching points.\nThe simplest treelike graph is a single tile with no interactions, and is evidently bound and deterministic.\nAdding a SIF tile (potentially SIF branching point) with a new interaction pair to this assembly graph cannot introduce a non-SIF branching point or cycle by definition.\nSince these are the only sources of nondeterminism and unboundedness, the assembly classification consequently cannot be altered.\nBy induction, any treelike graph is therefore bound and deterministic.\n\nAs such, any assembly graph that can be transformed into a treelike graph will also be bound and deterministic.\nWe now focus on the procedures of reducing an arbitrarily complex assembly graph to this deterministic state, or if this is not possible, identifying the source of nondeterminism or unboundedness.\n\nSequentially, we prune SIF tiles and check if the assembly graph is treelike.\nIf we cannot classify the assembly graph at this stage, we analyze the nature of any cycles present and remove trivial cycles, and examine the remaining cycles for unbound behavior.\nAlthough the graph approach identifies sources of nondeterminism and unboundedness, it is unable to predict spatial conflicts that lead to steric nondeterminism, a limitation discussed later.\n\n\\section{Alternative treelike assembly procedures}\\label{sec:ATAP}\n\nExplicitly transforming assembly graphs to be treelike is unnecessary; the steps of SIF tile pruning to remove trivial branching points and checking for infinite cycles is sufficient to determine the assembly classification.\n\nAssembly graphs which cannot undergo or fail the above procedures are necessarily nondeterministic or unbound.\nFor instance, any assembly graph with multiple of each complementary interface, i.e.\\ branching points connected to branching points, cannot be made treelike and thus is nondeterministic.\n\n\n\\subsection{SIF tiles elimination procedure} \\label{subsec:SEP}\nSIF tiles, by converse reasoning to that of the fundamental deterministic graph in Section \\ref{subsec:FDG}, can be `pruned' from an assembly graph without altering its assembly classification.\nThis is done by removing a SIF tile from the assembly graph, and neutralizing its complementary interface (or interfaces if it's a SIF branching point).\nSIF tiles whose complementary interface is a branching point are nondeterministic and may be classified without pruning.\n\nThis procedure is applied iteratively, and if a treelike graph is obtained after a removal, the assembly graph is known to be bound and deterministic.\nExamples of the procedure is shown in Figure \\ref{pruning}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.425\\textwidth]{SIF_elimination}\n \\caption{The SIF elimination procedure applied iteratively on two assembly graphs.\n The `1' SIF tile may be pruned in both (a) and (b), but only in (b) does the branching point become a SIF and allow further pruning.\n Note that the second stage of (b) is already treelike (no cycles or non-SIF branching points) and thus classified.\n Since (a) cannot be reduced to treelike, the assembly graph cannot be classified yet.}\\label{pruning}\n\\end{figure}\n\n\n\\subsection{Cycle rank classification}\n\nCycles can be categorized by the number of times the cycle pattern is repeated during the assembly, a quantity known as the rank.\nThe rank of the cycle can be determined from a single traversal of the cycle and noting the net rotation accumulated at the end of the walk.\nFigure \\ref{cycle_rank} shows several examples of cycles and their rank classification.\n\nFor square geometry, there are 4 periodic net rotations to consider, $\\theta=n \\pi \/ 2$ with $n \\in \\{-1,0,1,2\\}$. \nRank 4 cycles occur from $\\lvert n \\rvert=1$, while $n=2$ produces rank 2 cycles.\nThere are two subcategories for $n=0$, rank $\\infty$ and rank 1, based on spatial considerations.\n\nIf after traversing the cycle once, there is a zero net spatial translation, i.e.\\ the final tile placed reattaches in real-space to the first tile, then the cycle is rank 1.\nBy contrast, if there is a nonzero net spatial translation and the final tile placed leaves the final interface exposed, then the cycle will grow ad infinitum and is rank $\\infty$.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.475\\textwidth]{cycle_ranks}\n \\caption{The four cycle ranks possible with square geometry with assembly graphs on the left and assembled structures on the right in each example.\n (a) rank 2 intra-tile cycle ($\\Delta \\theta=\\pi$) (b) rank 4 inter-tile cycle ($\\Delta \\theta=-\\pi\/2$) (c) rank $\\infty$ inter-tile cycle ($\\Delta \\theta=\\infty, \\Delta_{XY} \\neq 0$) (d) rank 1 inter-tile cycle ($\\Delta \\theta=\\infty, \\Delta_{XY} = 0$).\n As soon as the initially placed tile is reused, the cycle rank may be classified, but for completeness the remaining assembly is shown in gray.\n}\\label{cycle_rank}\n\\end{figure}\n\n\n\\subsection{Establishing boundedness}\n\nCycles of rank 1, like SIFs, contribute trivial assembly behavior, and can be simplified without interfering with assembly classification.\nAny edge in the cycle may be removed, even if an edge is shared with another cycle.\nRegardless of the choice of cut, the resulting assemblies graphs will have the same quantity and rank of surviving cycles.\n\nThis procedure is likewise applied iteratively along with SIF elimination until the assembly graph is maximally simplified.\nMore details are given in Appendix \\ref{ap:cycle}.\nIf at any stage a rank $\\infty$ cycle is discovered, growth is immediately known to be unbound.\nMoreover, multiple surviving cycles ordinarily exhibit unboundness due to their amalgamated assembly pattern.\n\nSimplistically, this can be understood as the regeneration of cycle interfaces.\nWhen one cycle completes assembling, interfaces that assemble the other cycles are exposed and available for further growth.\nThis is demonstrated in Figure \\ref{pattern}.\n\nMultiple surviving cycles can have rank 1 behavior rather than rank $\\infty$, although it's rare due to the specific spatial constraints required.\nThis is detailed in Appendix \\ref{ap:cycle} for the specific case of Figure \\ref{pattern}.\n\nEstablishing the (in)finite behavior of the surviving cycles is hence nuanced, and proper care must be taken to identify infinite cycle behavior.\nFundamentally, infinite behavior can be identified if a tile in a cycle of finite rank $R$ is ever used in assembly more than $R$ times. \n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.375\\textwidth]{cycle_regen}\n \\caption{A section of an unbound structure resulting from an assembly graph with two rank 4 cycles.\nEvery time the blue (plain) cycle is completed, faces which interact with the red-hued (hatched) cycle are left exposed.\nA completion of the red-hued cycle likewise allows new copies of the blue cycle to start assembling, leading to unbound growth.}\\label{pattern}\n\\end{figure}\n\n\n\\section{Complete Procedure}\\label{sec:CP}\n\n\\subsection{Steric effects}\nIn the preceding sections we have addressed whether the assembly interactions alone make a tile set (non)deterministic or (un)bound.\nHowever, steric effects can also impact the growth of structures and cause nondeterminism and boundedness in assembly graphs that are otherwise deterministic or unbound.\n\nFor this reason a steric validation must be performed by growing the structure on the real-space lattice once.\nIf any lattice coordinate is used in assembly multiple times, then the assembly is sterically nondeterministic.\nSuch steric nondeterminism is clearly illustrated in Figure \\ref{mismatched}.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.35\\textwidth]{steric}\n \\caption{This assembly graph is rule-deterministic, but fails steric validation.\n Growing the structure on the real-space lattice reveals steric nondeterminism as the hatched lattice site does not have a unique occupant.\n This leads to nondeterminism as the final structure would depend on which occupant was built first.}\\label{mismatched}\n\\end{figure}\n\nSteric nondeterminism can prompt novel behavior in self-limiting cluster growth.\nIn previous experimental work \\cite{TES2016}, single-seed and multi-seed self-assembly were compared, observing that complementary pair interactions can limit growth in multi-seed assembly through local steric effects.\nSuch phenomenon is beyond the scope of this framework currently, but remains an topic for further analysis. \n\n\\subsection{Geometric symmetry}\nThe presence of symmetric tiles will never impact classifying a bound and deterministic assembly, but may mistake unbound growth for nondeterminism due to the symmetry-induced branching points.\nThese spurious nondeterministic branching points can be replaced with alternative interfaces that preserve assembly classification through a desymmetrization procedure (see Appendix \\ref{ap:sym} for details).\n\n\n\n\\subsection{Analysis Sequence}\n\nThe flowchart in Figure \\ref{flow} illustrates the steps in determining the (un)bound and (non)deterministic nature of a self-assembling tile set. \nAn open-source implementation of the assembly graph and steric algorithms is available online\\cite{LeonardGithub}. \n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{flowchart}\n \\caption{The sequential analysis of an assembly graph.\nClassifications can be bound and rule-deterministic (BD), unbound rule-deterministic (UD), or nondeterministic (ND).\nSteric validation must still be performed to ensure determinism.\nNote disconnected graphs are trivially seed-dependent, but each connected component may undergo this analysis.}\\label{flow}\n\\end{figure}\n\n\n\\section{Framework extensions}\\label{sec:FE}\nThe modular nature of the assembly graph algorithms allows the pinpointing of unbounded and nondeterministic behavior.\nRelaxing the conditions which detect the above classifications allows examining a wider range of assembly conditions and constraints.\n\n\\subsection{Other dimensions and geometries}\nThe procedures introduced here directly extend to regular triangular and hexagonal geometries, which allow regular tilings of the plane\\cite{1977}. \nThe only conceptual modification is the $\\theta$-shifts in cycle rank classification.\nTriangular geometry allows finite cycles of rank 1, 2, and 6, while hexagonal geometry allow ranks of 1, 2, 3, 4, and 6. \nSome cycles examples for regular hexagons are presented in Figure \\ref{exo}.\n\nThe procedures can also be extended to other dimensions, again only modifying cycle rank classification.\nFor instance, cubic geometry still only supports finite cycles ranks of 1, 2, and 4, while in one dimension only rank 2 finite cycles are possible.\nAlthough identifying infinite cycle behavior, steric validation, and other mentioned procedures are more complicated to implement, the concepts are unaltered.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.35\\textwidth]{hexagonal}\n \\caption{Hexagonal geometry allows for additional cycle ranks.\nA rank 3 and rank 2 cycle and the resulting structures are shown on the left and right respectively.}\\label{exo}\n\\end{figure}\n\n\\subsection{Nonstatic interaction rules}\nVarious biological assemblies do not the follow the earlier definition of determinism, due to the ability to form different final structures depending on external conditions, known as {\\em phenotype plasticity}.\nSuch external conditions can considerably impact self-assembly of biomolecules\\cite{deamer2006self,lavelle2009phase}.\nThis framework can encapsulate plasticity by considering e.g.\\ high pH interaction rules and low pH interaction rules, where the choice of rules determines which interactions are active.\nThis in turn can yield different deterministic structures.\n\nLikewise, it is possible to consider sequentially active interaction rules in a single assembly.\nAn assembly graph constructed under each set of interaction rules can be analyzed for (un)boundness and (non)determinism.\nThe assembly is then validated under steric constraints by assembling under the first set of interaction rules, followed by using the next set of rules on the existing structure etc.\nAs such, this approach can generalize to more complex assembly environments.\n\n\\subsection{Seed dependence}\nAlthough seed independence is ingrained in the method as outlined, incorporating fixed seed assembly allows for greater diversity in assembled structures.\nSeed dependence primarily results from branching points, as the behavior of SIF tiles and cycles are independent of the order of assembly.\n\nOne possible realization for modifying seed dependence is to walk on the assembly graph depth-first away from the seed.\nBranching points where a ``branched to'' face (i.e.\\ multiple faces with this interface type exist in the set) is discovered before the complementary ``branching'' face (i.e.\\ only face with this interface type in the set) can be ignored.\nIntuitively this removes the nondeterminism, as the assembly never encounters the multiple pathways when growing from those seeds.\nAn illustration of this extension is shown in Figure \\ref{seeded}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{seed_dependence}\n \\caption{An assembly graph that is classified as nondeterministic.\n However, if the assembly is seeded with tile type `2' or `3'\/`4', the structures are deterministic.\n The nondeterministic behavior of the branching point is only encountered walking from the `1' seed.}\\label{seeded}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:D}\n\n\n\\subsection{Advantages of the assembly graph framework}\nThe stochastic assembly method suffers from the probabilistic detection of unboundness and nondeterminism.\nClassification accuracy is directly related to the number of repeated assemblies, $K$, which compromises speed.\nFor example, stochastic assembly can miss the unbounded and nondeterministic nature of the tile set $\\{\\left(1,0,2,0\\right), \\left(1,2,0,0\\right)\\}$ even with seed independence.\nThe misclassification probability for this tile set has an analytic form for $K$ repeated assemblies given by\n\\[\n\\Pr = \\sum\\limits_{N=1}^{\\infty}[N (1\/2)^{3+2N}]^K\n\\]\nwhere $N$ is the number of pairs of the infinite cycle $\\left(1,0,2,0\\right)$ tiles in the assembly.\nFor $K=10$, the probability of misclassification is inconsequential ($10^{-15}$).\nHowever, more complex tile sets or weaker determinism definitions can lead to non-negligible loss of accuracy.\nAssembling $\\{\\left(1,2,0,0\\right), \\left(1,0,0,0\\right)\\}$ with the less restrictive shape determinism has a misclassification rate of $\\left(7\/8\\right)^{K\/2}$, over 50\\% for $K=10$.\n\nThe form of stochastic assembly algorithm (an example of which is sketched out in Section \\ref{intro}) also impacts probabilistic detection.\nIn the immediately preceding tile set example, randomly drawing a new tile gives equal probability to the two tile types.\nHowever, the $\\left(1,2,0,0\\right)$ tile type has two potential bindings, and so could also be considered to have twice the probability to be selected over the other tile type.\nDepending on the implementation choice, the misclassification rate varies between $\\left(19\/27\\right)^{K\/2}$ and $\\left(7\/8\\right)^{K\/2}$.\nThe graph method removes the need for external parameters and model specifications inherent in stochastic assembly.\n\nDue to the exponential growth of configuration space, only one, two, and three tile systems have been exhaustively searched.\nHowever, assembly graphs up to 20 tiles were tested with $10^{11}$ samples per graph size ($N_T$).\nAll sampled tile sets exhibited no misclassifications when compared with the ``truth'' outcomes of sufficiently reliable ($K=50 N_T$) stochastic assemblies.\nA direct comparison of the two methods' speeds is shown in Figure \\ref{fig:spm} for connected assembly graphs.\nDetails on the simulations used can be found in Appendix \\ref{ap:speed}.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.475\\textwidth]{speed_comparison}\n\n \\caption{Elapsed CPU time (s) per million connected assembly graphs classified for the graph approach and stochastic assembly with $K=10$.\nBoth methods initially scale with $N_T$ similarly, although with the graph approach approximately 3 times faster. \nHowever, for larger $N_T$ the stochastic method scales superlinearly while the graph framework scaling remains approximately linear.}\\label{fig:spm}\n\n\\end{figure}\n\n\n\\subsection{Nondeterministic assembly dynamics}\nIn \\cite{ARXIV2015}, nondeterministic self-assembly of single lattice tiles and mixtures of two tiles at varying relative concentrations were studied in detail.\nIn addition to using the conventional interaction rules introduced initially, assembly was also examined using {\\bf 1}s and {\\bf 2}s as self-interacting.\nTile sets were classified in terms of their behavior upon variation of tile density, recognizing critical transitions from bound to unbound growth and noncritical density transitions.\n\nThese dynamics can be analyzed from a different perspective, examining the 106 topologically distinct tile sets with the assembly graph framework.\nWhile identifying deterministic assemblies is relatively unchanged, the ability to robustly distinguish unbound from nondeterministic behavior is greatly improved using the assembly graph framework.\n \n\n\\subsection{Genotype-phenotype maps}\n\nThe lattice self-assembly model described here has been used as an abstract model of the genotype-phenotype (GP) map of protein quaternary structure \\cite{green}.\nIn these models, tile set is the genotype and the assembled structure represents the phenotype.\nThis work focused on bound deterministic assembly, while unbound or nondeterministic building block sets were regarded as a single unfavorable phenotype, and largely ignored.\n\nThe assembly graph framework allows assembly graphs to be used as an intermediate link between genotypes and phenotypes, and thus extend such work to examine the GP map of nondeterministic tile sets.\n\n\n\\subsection{Application to real proteins}\nThe study of such nondeterministic GP maps is essential, as nondeterministic self-assembly in the form of protein aggregation is a hallmark of numerous diseases.\nA classic example is that of hemoglobin and the sickle cell mutant.\n\nIn this representation, the $\\alpha$ and $\\beta$ chains of wild-type hemoglobin Hb A can be mapped to the $\\{ \\left(1, 3, 0, 0\\right), \\left(2, 0, 0, 4\\right)\\}$ tile set, and the sickle-cell mutant Hb S can be mapped to $\\{\\left(1, 3, 0, 0\\right),\\left(2, 5, 6, 4\\right)\\}$, displayed in Figure \\ref{GGP}.\nThe sickle-cell point mutation is sufficient to introduce an interaction labeled here with 5 and 6.\nThe wild-type assembly graph possesses a cycle of rank 2, and the mutation in the second tile introduces a new cycle of rank 4.\nFollowing the methodology we have outlined, this is sufficient to identify the unbound deterministic growth giving rise to the sickle cell anemia disease.\n \nBiological structures that are assembled through cycles with rank greater than 1 are prone to unbound growth via mutations which introduce additional cycles. \nProtein misfolding and resulting changes of interface angles in quaternary structures can be recast as introductions of cycles or branching points.\n\nWhile many proteins obviously exhibit more complex geometries than this framework is suited to, examining coarse-grained quaternary structure has already revealed insights into assembly properties and evolution history of proteins \\cite{Levy:2006ez,ahnert2015principles}.\nCombining these protein ``motifs'' with the introduced assembly graph formalisms is a rich vein for future exploration.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{sickle_cell}\n \\caption{The wild-type (left) and sickle-cell mutant (right) of the hemoglobin complex has been modelled previously using polyominoes\\cite{green}.\n The unbound growth denoted by arrows resulting from the mutation can now succinctly be pinpointed to the addition of a higher order cycle to the assembly graph.}\\label{GGP}\n\\end{figure}\n\n\\subsection{Universal computing}\nThe motivations behind this method are unassociated with those of universal computing and Turing machines, but there is conceptual overlap with other tile assembly models (TAMs).\nThe interaction rules used here define {\\em noncooperative} bindings, meaning the interaction is independent of factors beyond the two adjoined faces.\nSuch noncooperative models have been demonstrated to not intrinsically allow universal computing\\cite{meunier2017non}, requiring significant generalizations to the TAMs to increase their computational ability\\cite{hendricks2016doubles}.\n\nAs such, the framework outlined here for determining boundedness and determinism of self-assembling tile sets has little bearing on universal computation and the infamous halting problem.\n \n\\section{conclusion}\n\nWith the assembly graph framework we have introduced a new approach that can determine the (un)boundedness and (non)determinism of self-assembling tile sets.\nWhile the results presented have focused on 2D square geometry, the assembly framework readily extends to other dimensionalities (1D or 3D) and other geometries which regularly tile the plane. \n\nThe graph based approach outperforms the existing stochastic assembly approach in both speed and accuracy, facilitating the study of significantly larger GP maps and evolutionary dynamics.\nExternal parameter dependencies, like the $K$ repeated assemblies, are also eliminated.\n\nThe topological nature of the assembly graphs also opens new options for describing complexity and other properties relevant to evolutionary dynamics.\nIn addition, this methodology is a powerful tool for generalizing the study of GP maps by extending to the nondeterministic realm, which yields a coarse-grained model relevant to dysfunction and disease in the context of biological self-assembly.\n\n\\begin{acknowledgments}\nThis work was supported by the Engineering and Physical Sciences Research Council (ST and ASL), the Gatsby Foundation (ASL and SEA), and the Royal Society (SEA). \nThe authors would like to thank the referees for extensive feedback and suggestions on this work.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nUtterance and document level emotion recognition has received significant attention from the research community \\cite{SemEval2018Task1,poria2020beneath}. Given the utterance \\emph{Sudan protests: Outrage as troops open fire on protestors} an emotion recognition system will be able to detect that \\emph{anger} is the main expressed emotion, signaled by the word \"outrage\". However, the semantic information associated with expressions of emotion, such as the cause (the thing that triggers the emotion) or the target (the thing toward which the emotion is directed), is important to provide a finer-grained understanding of the text that might be needed in real-world applications. In the above utterance, the cause of the anger emotion is the event ``troops open fire on protestors'', while the target is the entity \"troops\" (see \\autoref{fig:goodnewseveryone}). \n\nResearch on finer-grained emotion analysis such as detecting the cause for an emotion expressed in text is in its infancy. Most work on emotion-cause detection has utilized a Chinese dataset where the cause is always syntactically realized as a clause and thus was modeled as a classification task \\citep{gui-etal-2016-event}. However, recently \\citet{DBLP:conf\/lrec\/BostanKK20} and \\citet{oberlander-klinger-2020-token} argued that in English, an emotion cause can be expressed syntactically as a clause (\\textit{as troops open fire on protestors}), noun phrase (\\textit{1,000 non-perishable food donations}) or verb phrase (\\textit{jumped into an ice-cold river}), and thus we follow their approach of framing emotion cause detection as a sequence tagging task. \n\nWe propose several ways in which to approach the tasks of emotion recognition and emotion cause tagging. First, these two tasks should not be independent; because the cause is the trigger for the emotion, knowledge about what the cause is should narrow down what emotion may be expressed, and vice versa. Therefore, we present a multi-task learning framework to model them jointly. Second, considering that common-sense knowledge plays an important role in understanding implicitly expressed emotions and the reasons for those emotions, we explore the use of common-sense knowledge via adapted knowledge models (COMET, \\citet{DBLP:conf\/acl\/BosselutRSMCC19}) for both tasks. A key feature of our approach is to combine these adapted knowledge models (i.e., COMET), which are specifically trained to use and express common-sense knowledge, with pre-trained language models such as BERT, \\citep{DBLP:conf\/naacl\/DevlinCLT19}. \n\nOur primary contributions are three-fold: (i) an under-studied formulation of the emotion cause detection problem as a sequence tagging problem; (ii) a set of models that perform the emotion classification and emotion cause tagging tasks jointly while using common-sense knowledge (\\autoref{sec:multitask}) with improved performance (\\autoref{sec:results}); and (iii) analysis to gain insight into both model performance and the GoodNewsEveryone dataset that we use \\citep{DBLP:conf\/lrec\/BostanKK20} (\\autoref{sec:analysis}).\n\n\\section{Related Work}\n\nEmotion detection is a widely studied subfield of natural language processing \\citep{SemEval2018Task1,poria2020beneath}, and has been applied to a variety of text genres such as fictional stories \\citep{10.3115\/1220575.1220648}, news headlines \\citep{DBLP:series\/sci\/StrapparavaM10}, and social media, especially microblogs such as Twitter \\citep{abdul-mageed-ungar-2017-emonet,10.5555\/2693068.2693087,DBLP:journals\/corr\/abs-1912-02387,SemEval2018Task1}. Earlier work, including some of the above, focused on feature-based machine learning models that could leverage emotion lexicons \\citep{DBLP:journals\/corr\/MohammadT13}), while recent work explores deep learning models (e.g., Bi-LSTM and BERT) and multi-task learning \\citep{DBLP:journals\/corr\/abs-1809-04505,demszky-etal-2020-goemotions}.\n\n\n\nHowever, comparatively few researchers have looked at the semantic roles related to emotion such as the cause, the target or the experiencer, with few exceptions for Chinese \\cite{gui-etal-2016-event,chen-etal-2018-joint,DBLP:journals\/corr\/abs-1906-01267,DBLP:journals\/corr\/abs-1906-01236,fan-etal-2020-transition,wei-etal-2020-effective,ding-etal-2020-ecpe}, English \\cite{mohammad-etal-2014-semantic,Ghazi2015DetectingES,kim-klinger-2018-feels,DBLP:conf\/lrec\/BostanKK20, oberlander-etal-2020-experiencers,oberlander-klinger-2020-token} and Italian \\cite{russo-etal-2011-emocause}. \nWe highlight some of these works here and draw connection to our work. Most recent work on emotion-cause detection has been carried out on a Chinese dataset compiled by \\citet{gui-etal-2016-event}. This dataset characterizes the emotion and cause detection problems as clause-level pair extraction problem -- i.e., of all the clauses in the input, one is selected to contain the expression of an emotion, and one or more (usually one) are selected to contain the cause of that emotion. Many publications have used this corpus to develop novel and effective model architectures for the clause-level classification problem \\citep{chen-etal-2018-joint,DBLP:journals\/corr\/abs-1906-01267,DBLP:journals\/corr\/abs-1906-01236,fan-etal-2020-transition,wei-etal-2020-effective,ding-etal-2020-ecpe}. The key difference between this work and ours is that we perform cause detection as a sequence-tagging problem: the cause may appear anywhere in the input, and may be expressed as any grammatical construction (a noun phrase, a verb phrase, or a clause). Moreover, we use common sense knowledge for both tasks (emotion and cause tagging), through the use of adapted language models such as COMET. \n\n\nFor English, several datasets have been introduced \\cite{mohammad-etal-2014-semantic, kim-klinger-2018-feels,Ghazi2015DetectingES,DBLP:conf\/lrec\/BostanKK20,poria2020recognizing}, and emotion cause detection has been tackled either as a classification problem \\cite{mohammad-etal-2014-semantic}, or as a sequence tagging or span detection problem \\cite{kim-klinger-2018-feels,Ghazi2015DetectingES,oberlander-klinger-2020-token,poria2020recognizing}. We particularly note the work of \\citet{oberlander-klinger-2020-token}, who argue for our problem formulation of cause detection as sequence tagging rather than as a classification task supported by empirical evidence on several datasets including the GoodNewsEveryone dataset \\cite{DBLP:conf\/lrec\/BostanKK20} we use in this paper. One contribution we bring compared to these models is that we formulate a multi-task learning framework to jointly learn the emotion and the cause span. Another contribution is the use of common-sense knowledge through the use of adapted knowledge models such as COMET (both in the single models and the multi-task models). \\newcite{ghosal-etal-2020-cosmic} have very recently shown the usefulness of common-sense reasoning to the task of conversational emotion detection. \n\n\\section{Data}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.5]{annotation_example_ours}\n\t\\end{center}\n\t\\caption{An example of the semantic roles annotated by \\citet{DBLP:conf\/lrec\/BostanKK20}}\n\t\\label{fig:goodnewseveryone}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[scale=0.5]{label_distribution}\n\t\\caption{Distribution of adjudicated emotion labels in the GoodNewsEveryone train data, as a percentage of the data points. ``Positive'' and ``Negative'' are abbreviated as + and -.}\n\t\\label{fig:label-distribution}\n\\end{figure}\n\n\\begin{figure*}\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\centering\n \\includegraphics[scale=0.35]{causefirst_nolstm_new.PNG}\n \\caption{The $\\text{Multi}_{C \\shortrightarrow E}$ model.} \\label{fig:causefirst-arch}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\centering\n \\includegraphics[scale=0.36]{emofirst_nolstm_new.PNG}\n \\caption{The $\\text{Multi}_{E \\shortrightarrow C}$ model.} \\label{fig:emofirst-arch}\n \\end{subfigure}\n \\caption{Our multi-task models.}\n\\end{figure*}\n\n\nFor our experiments, we use the GoodNewsEveryone corpus \\citep{DBLP:conf\/lrec\/BostanKK20}, which contains 5,000 news headlines labeled with emotions and semantic roles such as the target, experiencer, and cause of the emotion, as shown in \\autoref{fig:goodnewseveryone}.\\footnote{While the dataset labels both the most dominant emotion expressed in text and the reader's emotion, for this paper we only focus on the former.} \nWe focus on the emotion detection and cause tagging tasks in this work. To our knowledge, GoodNewsEveryone is the largest English dataset labeled for both of these tasks.\n\nIn our experiments, we limit ourselves to the data points for which a cause span was annotated (4,798). We also note that this dataset uses a 15-way emotion classification scheme, an extended set including the eight basic Plutchik emotions as well as additional emotions like \\textit{shame} and \\textit{optimism}. While a more fine-grained label set is useful for capturing subtle nuances of emotion, many external resources focus on a smaller set of emotions. We also note that the label distribution of this dataset heavily favors the more basic emotions, as shown in \\autoref{fig:label-distribution}. Therefore, for our work, we choose to limit ourselves to the six Ekman emotions (\\textit{anger}, \\textit{fear}, \\textit{disgust}, \\textit{joy}, \\textit{surprise}, and \\textit{sadness}). We also choose to keep \\textit{positive surprise} and \\textit{negative surprise} separated, to avoid severely unbalancing the label distribution for our experiments. We randomly split the remaining data (2,503 data points) into 80\\% train, 10\\% development, and 10\\% test.\n\n\\section{Models} \\label{sec:models}\n\nAn important feature showcased by the GoodNewsEveryone dataset is that causes of emotions can be expressed through different syntactic constituents such as clauses, verb phrases, or noun-phrases. Thus, we approach the cause detection problem as a sequence tagging problem using the IOB scheme \\citep{DBLP:journals\/corr\/cmp-lg-9505040}:\n$\\mathcal{C} = \\{\\text{I-cause}, \\text{O}, \\text{B-cause}\\}$. Our approach is supported by very recent results by \\citet{oberlander-klinger-2020-token} and \\citet{yuan-etal-2020-emotion} who show that modeling emotion cause detection as a sequence tagging problem is better suited than a clause classification problem, although not much current work has yet adopted this formulation. We tackle the emotion detection task as a seven-way classification task with $\\mathcal{E} = \\{\\text{anger}, \\text{disgust}, \\text{fear}, \\text{joy}, \\text{sadness}, \\text{negative}$ $\\text{surprise}, \\text{positive surprise}\\}$. \n\n\\subsection{Single-Task Models} \\label{sec:singletask}\n\nAs a baseline, we train single-task models for each of emotion classification and cause span tagging. We use a pre-trained BERT language model\\footnote{We use \\textsc{BERT-base-uncased}. We experimented with \\textsc{BERT-base-cased}, but it underperformed as the headlines incorporated into GoodNewsEveryone come from different news sources and have different capitalization styles.} \\citep{DBLP:conf\/naacl\/DevlinCLT19}, which we fine-tune on our data, as the basis of this model. Our preprocessing strategy for all of our models consists of the pretrained BERT vocabulary and WordPiece tokenizer\\footnote{In the tagging setting, we ignore all tags predicted for subword tokens and use only the tag of the first subword.} \\citep{DBLP:journals\/corr\/WuSCLNMKCGMKSJL16} from Huggingface \\citep{wolf-etal-2020-transformers}. Therefore, for a sequence of $n$ WordPiece tokens, our input to the BERT model is a sequence of $n + 2$ tokens, $X = [[\\text{CLS}], x_1, x_2, ... x_n, [\\text{SEP}]]$, where each $x_i$ is from a finite WordPiece vocabulary and [CLS] and [SEP] are BERT's begin and end tokens. Passing $X$ through BERT yields a sequence of vector hidden states $H = [h_{[CLS]}, h_1, h_2, ..., h_n, h_{[SEP]}]$ with dimension $d_{BERT} = 768$. For emotion classification, we pool these hidden states and allow hyperparameter tuning to select the best type: selecting the [CLS] token ($h_f = h_{[CLS]}$), mean pooling ($h_f = \\frac{\\sum_{i=1}^n h_i}{n}$), max pooling ($h_{f,j} = \\max{h_{i,j}}$), or attention as formulated by \\citet{DBLP:journals\/corr\/BahdanauCB14}:\n\n\\begin{equation} \\label{eqn:bahdanau-attn}\n\th_f = \\sum^{n}_{i=1} \\alpha_i h_i\n\\end{equation}\n\nwhere $\\alpha_i = \\frac{\\exp{(W_ah_i + b_a)}}{\\sum_{j=1}^n \\exp{(W_ah_j + b_a)}}$ for trainable weights $W_a \\in \\mathbb{R}^{1 \\times d_{BERT}}$ and $b_a \\in \\mathbb{R}^{1}$. Then, the final distribution of emotion scores is calculated by a single dense layer and a softmax: \n\n\\begin{equation} \\label{eqn:emotion}\n\te = \\text{softmax}(W_eh_f + b_e)\n\\end{equation}\n\nwith $e \\in \\mathbb{R}^{|\\mathcal{E}|}$ and for trainable parameters $W_e \\in \\mathbb{R}^{|\\mathcal{E}| \\times d_{BERT}}$ and $b_e \\in \\mathbb{R}^{|\\mathcal{E}|}$. For cause tagging, a tag probability distribution is calculated directly on each hidden state: \n\n\\begin{equation} \\label{eqn:cause}\n\tc_i = \\text{softmax} (W_ch_i + b_c)\n\\end{equation}\n\nwith $c_i \\in \\mathbb{R}^{|\\mathcal{C}|}$ and for trainable parameters $W_c \\in \\mathbb{R}^{|\\mathcal{C}| \\times d_{BERT}}$ and $b_c \\in \\mathbb{R}^{|\\mathcal{C}|}$. We refer to both of these single-task models as BERT; if the task is not clear from the context, we will refer to the emotion detection model as $\\text{BERT}_E$ and the cause tagging model as $\\text{BERT}_C$. Our training loss for emotion classification as well as emotion cause tagging is the mean negative log-likelihood (NLL) loss per minibatch of size \\textit{b}:\n\n\\begin{equation} \\label{eqn:emo-loss}\n\t\\text{NLL}_{\\text{emo}} = - \\frac{1}{b} \\sum_j \\sum_k y_{jk} \\log e_{jk}\n\\end{equation}\n\n\\begin{equation} \\label{eqn:cause-loss}\n\t\\text{NLL}_{\\text{cause}} = - \\frac{1}{b} \\sum_i \\sum_j \\sum_k y_{ijk} \\log c_{ijk}\n\\end{equation}\n\nwhere $j$ is the index of the sentence in the minibatch, $k$ is the index of the label being considered (emotion labels for $\\text{NLL}_{\\text{emo}}$ and IOB tags for $\\text{NLL}_{\\text{cause}}$), $i$ is the index of the $i^{th}$ token in the $j^{th}$ sentence in the minibatch, $y_{jk} \\in \\{0, 1\\}$ is the gold probability of the $k^{th}$ emotion label for the $j^{th}$ sentence, $y_{ijk} \\in \\{0, 1\\}$ is the gold probability of the $k^{th}$ cause tag for the $i^{th}$ token in the $j^{th}$ sentence, and $e_{jk}$ and $c_{ijk}$ are the output probabilities of the ${k^{th}}$ emotion label and of the $k^{th}$ cause label for the $i^{th}$ token, both for the $j^{th}$ sentence.\n\n\\subsection{Multi-Task Models} \\label{sec:multitask}\n\nOur hypothesis is that the emotion detection and cause tagging tasks are closely related and can inform each other; therefore we propose three multi-task learning models to test this hypothesis. For all multi-task models, we use the same base architecture (BERT) as the single models. Additionally, for these models, we combine the losses of both tasks and weight them with a tunable lambda parameter: $\\lambda \\text{NLL}_\\text{emo} + (1 - \\lambda) \\text{NLL}_\\text{cause}$, using $\\text{NLL}_\\text{emo}$ and $\\text{NLL}_\\text{cause}$ from \\autoref{eqn:emo-loss} and \\autoref{eqn:cause-loss}.\n\n\\paragraph{Multi.} The first model, $\\text{Multi}$, is the classical multi-task learning framework with hard parameter sharing, where both tasks share the same BERT layers. Two dense layers for emotion classification and cause tagging operate at the same time from the same BERT layers, and we train both of the tasks simultaneously. That is, we simply calculate our emotion scores $e$ and cause tag scores $c$ from the same set of hidden states $H$.\n\nWe further develop two additional multi-task models with the intuition that we can design more explicit and concrete task dependencies than simple parameter sharing in the representation layer.\n\n\\paragraph{$\\text{Multi}_{C \\shortrightarrow E}$.} We assume that if a certain text span is given as the cause of an emotion, it should be possible to classify that emotion correctly while looking only at the words of the cause span. Therefore, we propose the $\\text{Multi}_{C \\shortrightarrow E}$ model, the architecture of which is illustrated in \\autoref{fig:causefirst-arch}. This model begins with the single-task cause detection model, $BERT_C$, which produces a probability distribution $P(y_i | x_i)$ over IOB tags for each token $x_i$, where $P(y_i | x_i) = c_i$ from \\autoref{eqn:cause}. Then, for each token, we calculate the probability that it is part of the cause as $P(\\text{Cause} | x_i) = P(B | x_i) + P(I | x_i) = 1 - P(O | x_i)$. We feed the resulting probabilities through a softmax over the sequence and use them as an attention distribution over the input tokens in order to pool the hidden representations and perform emotion classification: attention is computed as in \\autoref{eqn:bahdanau-attn}, where $\\alpha_i = \\frac{\\exp{P(\\text{Cause} | x_i)}}{\\sum_{j=1}^n \\exp{P(\\text{Cause} | x_i)}}$, and emotion classification as in \\autoref{eqn:emotion}. For the $\\text{Multi}_{C \\shortrightarrow E}$ model, we apply teacher forcing at training time, and the gold cause spans are used to create the attention weights before emotion classification (which means that $P(\\text{Cause} | x_i) \\in \\{0, 1\\}$). At inference time, the model uses the predicted cause span instead. \n\n\\paragraph{$\\text{Multi}_{E \\shortrightarrow C}$.} Next, we hypothesize that knowledge of the predicted emotion should help us identify salient cause words. The $\\text{Multi}_{E \\shortrightarrow C}$ model first performs emotion classification, which results in a probability distribution over predicted emotion labels, as in the $\\text{BERT}_E$ model and \\autoref{eqn:emotion}. We additionally keep an emotion embedding matrix $E$, where $E[i]$ is a learnable representation of the $i$-th emotion label (see \\autoref{fig:emofirst-arch}) with dimension $d_e$ (in our experiments, we set $d_e = 300$). We use the predicted label probabilities $e$ to calculate a weighted sum of the emotion embeddings, i.e., $M = \\sum_i e_i \\cdot E[i]$. We then concatenate $M$ to the hidden representation of each token and perform emotion cause tagging with a final dense layer, i.e., $c_i = \\text{softmax} (W_{c'}[h_i ; M] + b_{c'})$, where $;$ is the concatenation operator and $W_{c'} \\in \\mathbb{R}^{|\\mathcal{C}| \\times (d_{BERT} + d_e)}$ and $b_{c'} \\in \\mathbb{R}^{|\\mathcal{C}|}$ are trainable parameters. In the $\\text{Multi}_{E \\shortrightarrow C}$ model, we again do teacher forcing and use the gold emotion labels before doing the sequence tagging for cause detection (i.e., $e$ is a one-hot vector where the gold emotion label has probability 1 and all other emotion labels have probability 0). At inference time, the model will use the predicted emotion distribution instead.\n\n\\subsection{Adapted Knowledge Models}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[scale=0.35]{comet_causefirst_nolstm_new.PNG}\n\t\\caption{The architecture of our proposed $\\text{Multi}^{COMET}_{C \\shortrightarrow E}$ model.}\n\t\\label{fig:comet-causefirst-architecture}\n\\end{figure}\n\n\\begin{table*}[t!]\n\t\\centering\n\t\\begin{tabular}{l|r|r|r}\n\t\t& \\textbf{Emotion Macro F1} & \\textbf{Emotion Accuracy} & \\textbf{Cause Span F1} \\\\ \\hline\n\t\tBERT & 37.25 $\\pm$ 1.30 & 38.50 $\\pm$ 0.84 & 37.49 $\\pm$ 1.94 \\\\ \\hline \n\t\t$\\text{BERT}^{COMET}$ & 37.74 $\\pm$ 0.84 & 38.50 $\\pm$ 1.14 & 39.27 $\\pm$ 1.85 \\\\ \\hline\n\t\tMulti & 36.91 $\\pm$ 1.48 & 38.34 $\\pm$ 1.94 & 38.35 $\\pm$ 3.89 \\\\ \\hline\n\t\t$\\text{Multi}_{C \\shortrightarrow E}$ & 37.74 $\\pm$ 2.12 & 38.74 $\\pm$ 2.07 & 39.08 $\\pm$ 3.73 \\\\ \\hline\n\t\t$\\text{Multi}_{E \\shortrightarrow C}$ & 38.26 $\\pm$ 3.28 & 39.69 $\\pm$ 3.41 & 38.83 $\\pm$ 1.60 \\\\ \\hline\n\t\t$\\text{Multi}^{COMET}$ & 37.06 $\\pm$ 2.04 & 39.05 $\\pm$ 0.98 & \\textbf{39.50} $\\pm$ 2.25 \\\\ \\hline\n\t\t$\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ & \\textbf{39.26}* $\\pm$ 1.13 & \\textbf{40.79} $\\pm$ 2.17 & 38.68 $\\pm$ 1.36 \\\\ \\hline\n\t\t$\\text{Multi}_{E \\shortrightarrow C}^{COMET}$ & 37.44 $\\pm$ 1.37 & 38.58 $\\pm$ 1.44 & 36.27 $\\pm$ 1.31 \\\\\n\t\\end{tabular}\n\t\\caption{The results of our models, averaged over five runs with the same five distinct random seeds. The model with the highest mean performance under each metric is bolded. Results marked with a * are statistically significant above the single-task BERT baseline by the paired t-test (p $<$ 0.05).}\n\t\\label{tab:results}\n\\end{table*}\n\nRecent work has shown that fine-tuning pre-trained language models such as GPT-2 on \\emph{knowledge graph tuples} such as ConceptNet~\\cite{li-etal-2016-commonsense} or ATOMIC \\citep{DBLP:journals\/corr\/abs-1811-00146} allows these models to express their implicit knowledge directly \\citep{DBLP:conf\/acl\/BosselutRSMCC19}. These adapted \\emph{knowledge models} (e.g., COMET \\citep{DBLP:conf\/acl\/BosselutRSMCC19}) can produce common-sense knowledge on-demand for any entity, relation or event.\nConsidering that common-sense knowledge plays an important role in understanding implicitly expressed emotions and the reasons for those emotions, we explore the use of common-sense knowledge for our tasks, in particular the use of COMET adaptively pre-trained on the ATOMIC event-centric knowledge base. ATOMIC's event relations include ``xReact'' and ``oReact'', which describe the feelings of certain entities after the input event occurs. For example, ATOMIC's authors present the example of $<$PersonX pays PersonY a compliment, xReact, PersonX will feel good$>$. xReact refers to the feelings of the primary entity in the event, and oReact refers to the feelings of others (in this instance, oReact yields ``PersonY will feel flattered''). For example, using the headline ``Sudan protests: Outrage as troops open fire on protestors\", COMET-ATOMIC outputs that PersonX feels justified, PersonX feels angry, Others feel angry, and so on (\\autoref{fig:comet-causefirst-architecture}). To use this knowledge model in our task, we modify our approach by reframing our single-sequence classification task as a sequence-pair classification task (for which BERT can be used directly). We feed our input headlines into COMET-ATOMIC (using the model weights released by the authors), collect the top two outputs for xReact and oReact using beam search decoding, and then feed them into BERT alongside the input headlines, as a second sequence using the SEP token. That is, our input to BERT is now $X = [[\\text{CLS}], x_1, x_2, ..., x_n, [\\text{SEP}], z_1, z_2, ..., z_m, [\\text{SEP}]]$, where $z_i$ are the $m$ WordPiece tokens of our COMET output and are preprocessed in the same way as $x_i$. We hypothesize that, since pre-trained BERT is trained with a next sentence prediction objective, expressing the COMET outputs as a grammatical sentence will help BERT make better use of them, so we formulate this second sequence as complete sentences (e.g., ``This person feels... Others feel...'') (\\autoref{fig:comet-causefirst-architecture}). \n \nThis approach allows us incorporate information from COMET into all our single- and multi-task BERT-based models; the example shown in \\autoref{fig:comet-causefirst-architecture} is our $\\text{Multi}_{C \\shortrightarrow E}$ model. We refer to the COMET variants of these models as: $\\text{BERT}^{COMET}$ (single-task models) and $\\text{Multi}^{COMET}$, $\\text{Multi}^{COMET}_{C \\shortrightarrow E}$, $\\text{Multi}^{COMET}_{E \\shortrightarrow C}$ for the three multi-task models.\n\n\\section{Experimental Setup}\n\n\\paragraph{Evaluation Metrics}\nFor emotion classification, we report macro-averaged F1 and accuracy. For cause tagging, we report exact span-level F1 (which we refer to as \\textit{span F1}), as developed for named entity recognition (e.g., \\citet{tjong-kim-sang-de-meulder-2003-introduction}), where a span is marked as correct if and only if its type and span boundaries match the gold exactly\\footnote{Our cause tagging task has only one type, ``cause'', as GoodNewsEveryone is aggregated such that each data point has exactly one emotion-cause pair. We note that this problem formulation leaves open the possibility of multiple emotion-cause pairs.}.\n\n\\paragraph{Training and Hyperparameter Selection} The classification layers are initialized randomly from a uniform distribution over $[-0.07,0.07]$\\footnote{The default initialization from the \\texttt{gluon} package: \\url{https:\/\/mxnet.apache.org\/versions\/1.7.0\/api\/python\/docs\/api\/gluon\/index.html}}, and all the parameters are trained on our dataset for up to 20 epochs, with early stopping based on the performance on the validation data (macro F1 for emotion, span F1 for cause). All models are trained with the Adam optimizer \\citep{DBLP:journals\/corr\/KingmaB14}. We highlight again that for our $\\text{Multi}_{C \\shortrightarrow E}$ and $\\text{Multi}_{E \\shortrightarrow C}$ models, we use teacher forced during training to avoid cascading training error. Because the subset of the data we use is relatively small, we follow current best practices for dealing with neural models on small data and select hyperparameters and models using the average performance of five models with different fixed random seeds on the development set. We then base our models' performance on the average of the results from these five runs (e.g., reported emotion F1 is the average of the emotion F1 scores for each of our five runs). For our joint models, since our novel models revolve around using one task as input for the other, we separately tune two sets of hyperparameters for each model, one based on each of the single-task metrics, yielding, for example, one Multi model optimized for predicting emotion and one optimized for predicting cause. The hyperparameters we tune are dropout in our linear layers, initial learning rate of the optimizer, COMET relation type, lambda weight for our multi-task models, and the type of pooler for emotion classification (enumerated in \\autoref{sec:singletask}).\n\n\\section{Results} \\label{sec:results}\n\nWe present the results of our models in \\autoref{tab:results}\\footnote{\\citet{oberlander-klinger-2020-token} report an F1 score of 34 in this problem setting on this dataset, but on a larger subset of the data (as they do not limit themselves to the Ekman emotions) and so we cannot directly compare our work to theirs.}. We see that the overall best model for each task is a multi-task adapted knowledge model, with $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ performing best for emotion (which is a statistically significant improvement over BERT by the paired t-test, $p<0.05$) and $\\text{Multi}^{COMET}$ performing best for cause. These results seem to support our two hypotheses: 1) emotion recognition and emotion cause detection can inform each other and 2) common-sense knowledge is helpful to infer the emotion and the cause for that emotion expressed in text.\nSpecifically, we notice that $\\text{Multi}_{C \\shortrightarrow E}$ alone does not outperform BERT on either cause or emotion, but $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ outperforms both BERT and $\\text{Multi}_{C \\shortrightarrow E}$ on both tasks. For cause, we also see additional benefits of common-sense reasoning alone: $\\text{BERT}^{COMET}$ outperforms BERT (multi-task modeling alone, Multi, also outperforms BERT for this task) and $\\text{Multi}^{COMET}$ outperforms Multi. These results speak to the differences between the two tasks, suggesting that common-sense reasoning, which aims to generates implicit emotions, and cause information may be complementary for emotion detection, but that for cause tagging, common-sense reasoning and given emotion information may overlap. \nThe common-sense reasoning we have used in this task (xReact and oReact from ATOMIC) is expressed as possible emotional reactions to an input situation, so this makes intuitive sense.\n\n\\begin{figure}[t]\n \\includegraphics[scale=0.55]{emotion_comparison_new.png}\n \\caption{Performance of the BERT and $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ models on emotion classification.}\n \\label{fig:emotion_breakdown}\n\\end{figure}\n\nFinally, we also present per-emotion results for our best model for each task ($\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ for emotion and $\\text{Multi}^{COMET}$ for cause) against the single-task BERT baselines in \\autoref{fig:emotion_breakdown} and \\autoref{fig:cause_breakdown}; these per-emotion scores are again the average performance of models trained with each of our five random seeds. We see that each task improves on a different set of emotions: for emotion classification $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ consistently improves over BERT by a significant margin on joy and to a lesser extent on anger and sadness. Meanwhile, for cause tagging, $\\text{Multi}^{COMET}$ improves over BERT on anger, disgust, and fear, while yielding very similar performance on the rest of the emotions.\n\n\\section{Analysis and Discussion} \\label{sec:analysis}\n\nIn order to understand the impact of common-sense reasoning and multi-task modeling for the two tasks, we provide several types of analysis in addition to our results in \\autoref{sec:results}. First, we include examples of our various models' outputs showcasing the impact of our methods (\\autoref{sec:examples}). Second, we carry out an analysis of the dataset, focusing on the impact of label variation among multiple annotators on the models' performance (\\autoref{sec:label-analysis}).\n\n\\subsection{Example Outputs} \\label{sec:examples}\n\\begin{table*}[t]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t\\textbf{BERT} & \\textbf{Multitask} & \\textbf{$\\text{BERT}^{COMET}$} & \\textbf{$\\text{Multitask}^{COMET}$} \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Mexico reels from \\hlcolor{shooting attack in El Paso}\\\\ \\textbf{\\textit{fear}}\\end{tabular}} \\\\ \\hline\n\t\tnegative surprise & negative surprise & fear & fear \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Insane video shows Viking Sky cruise \\hlcolor{ship thrown into chaos at sea}\\\\ \\textbf{\\textit{fear}}\\end{tabular}} \\\\ \\hline\n\t\tnegative surprise & fear & negative surprise & fear \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Durant \\hlcolor{could return for Game 3}\\\\ \\textbf{\\textit{positive surprise}}\\end{tabular}} \\\\ \\hline\n\t\tfor game & \\multicolumn{3}{|c|}{could return for game} \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Dan Fagan: \\hlcolor{Triple shooting near New Orleans School yet another sign of city's crime problem}\\\\ \\textbf{\\textit{negative surprise}}\\end{tabular}} \\\\ \\hline\n\t\t\\begin{tabular}[c]{@{}c@{}}school yet another sign \\\\ of city's crime\\end{tabular} & \\multicolumn{3}{|c|}{\\begin{tabular}[c]{@{}c@{}}: triple shooting near new orleans school yet another sign of city's \\\\ crime\\end{tabular}} \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Example outputs from our systems. For each example, the gold cause is highlighted in yellow and the gold emotion is given under the text; the first two examples give our models' emotion outputs; the latter two, their causes. Joined cells show that multiple models produced the same output. To make this table easier to read, ``Multitask'' here may refer to Multi, $\\text{Multi}_{E \\shortrightarrow C}$, or $\\text{Multi}_{C \\shortrightarrow E}$ (details on selection and results for each individual model available in appendix; most multi-task models gave similar outputs).}\n\t\\label{tab:examples}\n\\end{table*}\n\n\\begin{table*}[t]\n\t\\begin{adjustbox}{max width=\\textwidth}\n\t\t\\begin{tabular}{c|l|l|l|l|l|l|l|l}\n\t\t\t\\textbf{Metric} & \\textbf{BERT} & \\textbf{$\\text{BERT}^{COMET}$} & \\textbf{Multi} & \\textbf{$\\text{Multi}_{E \\shortrightarrow C}$} & \\textbf{$\\text{Multi}_{C \\shortrightarrow E}$} & \\textbf{$\\text{Multi}^{COM}$} & \\textbf{$\\text{Multi}_{E \\shortrightarrow C}^{COMET}$} & \\textbf{$\\text{Multi}_{C \\shortrightarrow E}^{COMET}$} \\\\ \\hline\n\t\t\t\\begin{tabular}[c]{@{}l@{}}\\textbf{Acc.}\\\\\\textbf{(Gold)}\\end{tabular} & 38.50 & 38.50 & 38.34 & 39.68 & 38.74 & 39.05 & 38.58 & 40.79 \\\\ \\hline\n\t\t\t\\begin{tabular}[c]{@{}l@{}}\\textbf{Acc.}\\\\\\textbf{($\\neg$Gold)}\\end{tabular} & 23.48 & 23.24 & 22.37 & 21.11 & 22.85 & 21.26 & 22.45 & 20.08\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Comparison of gold accuracy and non-gold ($\\neg$gold) accuracy for our emotion classification models.}\n\t\\label{tab:annotator-comparison}\n\\end{table*}\n\nWe provide some example outputs from our systems for both cause and emotion in \\autoref{tab:examples}; the various Multi models have been grouped together for readability and because they often produce similar outputs, but the outputs for every model are available in the appendix. In the first example, the addition of COMET to BERT informs the model enough to choose the gold emotion label; in the third and fourth, either COMET or multi-task learning is enough to help the model select key words that should be included in the cause (\\textit{return} and \\textit{triple shooting}). We also particularly note the second example, in which multi-task learning is needed both for the BERT and $\\text{BERT}^{COMET}$ models to be able to correctly predict the gold emotion. This suggests that for cause, both common-sense reasoning and emotion classification may carry overlapping useful information for cause tagging, while for emotion, different instances may be helped more by different aspects of our models.\n\n\n\\subsection{Label Agreement} \\label{sec:label-analysis}\n\n\\begin{figure}[t]\n\t\\includegraphics[scale=0.55]{cause_comparison_new.png}\n\t\\caption{Performance of the BERT and $\\text{Multi}^{COMET}$ models on cause tagging, broken down by emotion.}\n\t\\label{fig:cause_breakdown}\n\\end{figure}\n\nUpon inspection of the GoodNewsEveryone data, we discover significant variation in the emotion labels produced by annotators as cautioned by the authors in their original publication\\footnote{While the authors selected data according to agreement on the emotion labeling task, they found that in only 75\\% of cases do at least 3 annotators agree, with diminishing rates of agreement for more annotators.}. \nFrom our inspection of the development data, we see recurring cases where different annotators give directly opposing labels for the same input, depending on how they interpret the headline and whose emotions they choose to focus on. For example, our development set includes the following example: \\textit{Simona Stuns Serena at Wimbledon: Game, Set and ``Best Match'' for Halep}. The gold adjudicated emotion label for this example is \\textit{negative surprise}, but annotators actually included multiple primary and secondary emotion labels including \\textit{joy}, \\textit{negative surprise}, \\textit{positive surprise}, \\textit{pride}, and \\textit{shame}, which can be understood as various emotions felt by the two entities participant in the event (Simona Halep and Serena Williams). For this input, COMET suggests xReact may be \\textit{happy} or \\textit{proud} and oReact may be \\textit{happy} --- these reactions are likely most appropriate for tennis player Simona Halep, but not the only possible emotion that can be inferred from the headline.\n\nInspired by the variation in the data, we compute also models' accuracy using the human annotations that did not agree with the gold (i.e., a predicted emotion label is correct if it was suggested by a human annotator but was not part of a majority vote to be included in the gold). We denote this $\\neg$Gold, and we compare the performance of our models with respect to Gold and $\\neg$Gold. We present the results of this analysis in \\autoref{tab:annotator-comparison}\\footnote{Note that we perform this analysis on just one of our five runs of the model, so the accuracy numbers do not exactly correspond to those in \\autoref{tab:results}.}. \nIn this table, a higher $\\neg$Gold accuracy means that the model is more likely to produce emotion labels that were not the gold but were suggested by some annotator. First of all, we note that all models have a relatively high $\\neg$Gold accuracy (about half the magnitude of their gold accuracy); we believe this reflects the wide variety of annotations given by the annotators. We see a tradeoff between the Gold and $\\neg$Gold accuracy, and we note that generally the single-task models have higher $\\neg$Gold accuracy and the COMET-enhanced multi-task models have higher Gold accuracy. This suggests that our language models have general knowledge about emotion already, but that applying common-sense knowledge helps pare down the space of plausible outputs to those that are most commonly selected by human annotators. Recall that this dataset was annotated by taking the most frequent of the annotator-provided emotion labels.\nFurther, since the multi-task models have higher Gold accuracy and lower $\\neg$Gold accuracy than the single-task models, this suggests that also predicting the cause of an emotion causes the model to narrow down the space of possible emotion labels to only those that are most common.\n\n\\section{Conclusions and Future Work}\n\nWe present a common-sense knowledge-enhanced multi-task framework for joint emotion detection and emotion cause tagging. Our inclusion of common-sense reasoning through COMET, combined with multi-task learning, yields performance gains on both tasks including significant gains on emotion classification.\nWe highlight the fact that this work frames the cause extraction task as a span tagging task, allowing for the future possibility of including multiple emotion-cause pairs per input or multiple causes per emotion and allowing the cause to take on any grammatical role. Finally, we present an analysis of our dataset and models, showing that labeling emotion and its semantic roles is a hard task with annotator variability,\nbut that common-sense knowledge helps language models focus on the most prominent emotions according to human annotators. In future work, we hope to explore ways to integrate common-sense knowledge more innately into our classifiers and ways to apply these models to other fine-grained emotion tasks such as detecting the experiencer or the target of an emotion.\n\n\\section*{Acknowledgements}\n\nWe would like to thank our reviewers as well as the members of Amazon AI team for their constructive and insightful feedback.\n\n\n\\section*{Ethical Considerations}\n\nOur intended use for this work is as a tool to help understand emotions expressed in text. We propose that it may be useful for things like product reviews (where producers and consumers can rapidly assess reviews for aspects of their products to improve or expand), disaster relief (where those in need of help from any type of disaster can benefit if relief agents can understand what events are causing negative emotions, during and after the initial disaster), and policymaking (where constituents can benefit if policymakers can see real data about what policies are helpful or not and act in their interests). These applications do depend on the intentions of the user, and a malicious actor may certainly misuse the ability to (accurately or inaccurately) detect emotions and their causes. We do not feel it responsible to publicly list the ways in which this may happen in this paper. We also believe that regulators and operators of this technology should be aware that it is still in its nascent stages and does not represent an infallible oracle --- the predictions of this and any model should be reviewed by humans in the loop, and we feel that general public awareness of the limitations and mistakes of these models may help mitigate any possible harm. If these models are inaccurate, they will output either the incorrect emotion or the incorrect cause; blindly trusting the model's predictions without examining them may lead to unfair consequences in any of the above applications (e.g., failure to help someone whose text is misclassified as positive surprise during a natural disaster or a worsened product or policy if causes are incorrectly predicted). We additionally note that in its current form, this work is intended to detect the emotions that are expressed in text (headlines), and not those of the reader.\n\n\nWe concede that the data used in this work consists of news headlines and may not be the most adaptable to the use cases we describe above; we caution that models trained on these data will likely require domain adaptation to perform well in other settings. \\citet{DBLP:conf\/lrec\/BostanKK20} report that their data comes from the Media Bias Chart\\footnote{\\url{https:\/\/www.adfontesmedia.com\/about-the-interactive-media-bias-chart\/}}, which reports that their news sources contain a mix of political views, rated by annotators who also self-reported a mix of political views. We note that these data are all United States-based and in English. \\citet{DBLP:conf\/lrec\/BostanKK20} do sub-select the news articles according to impact on Twitter and Reddit, which have their own user-base biases\\footnote{\\url{https:\/\/www.pewresearch.org\/internet\/fact-sheet\/social-media\/}}, typically towards young, white American men; therefore, the data is more likely to be relevant to these demographics. The language used in headlines will likely most resemble Standard American English as well, and therefore our models will be difficult to use directly on other dialects and vernaculars.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}