Content-Type: multipart/mixed; boundary="-------------0204092314929" This is a multi-part message in MIME format. ---------------0204092314929 Content-Type: text/plain; name="02-178.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-178.keywords" infinite dimensional symplectic groups, CCR algebra, quasifree state ---------------0204092314929 Content-Type: application/x-tex; name="shimada.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="shimada.tex" %% Do not edit this file unless you know what you are doing. \documentclass[12pt,a4paper]{article} %\usepackage[T1]{fontenc} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. \providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@} \newcommand{\noun}[1]{\textsc{#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. %\usepackage[T1]{fontenc} \makeatletter %%%%%% package %%%%%% \usepackage{latexsym} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{amsthm} \usepackage{amscd} \usepackage{plext} \usepackage[dvips]{graphicx} \usepackage{enumerate} %%%%%% layout %%%%%%%%% \setlength{\textheight}{24.0cm} \setlength{\textwidth}{16.0cm} %%\parindent=4.5mm \setlength{\topmargin}{-1.0cm} \setlength{\evensidemargin}{0cm} \setlength{\oddsidemargin}{0cm} %%%%%% definition,theorem,etc... %%%%%% \swapnumbers \theoremstyle{definition} \newtheorem{thm}{Theorem}[section] \newtheorem{defini}[thm]{Definition} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{asp}[thm]{Assumption} \newtheorem{ex}[thm]{Example} \newtheorem{lm}[thm]{Lemma} \newtheorem{remark}[thm]{Remark} %%%%%% number of equation %%%%%% \makeatletter \renewcommand{\theequation}{ \thesection.\arabic{equation}} \@addtoreset{equation}{section} \makeatother %%%%%% Environment of ``proof'' %%%%%%%% \renewcommand{\qed}{ \hfill \hbox{\rule[-2pt]{3pt}{6pt}}} %%%%%% others %%%%%% \renewcommand{\labelenumi}{(\theenumi)} \renewcommand{\theenumii}{\roman{enumii}} \renewcommand{\labelenumii}{(\theenumii)} \def\rnum#1{\expandafter{\romannumeral #1}} \def\Rnum#1{\uppercase\expandafter{\romannumeral #1}} \makeatother \begin{document} \newpage\thispagestyle{empty} {\topskip 2cm \begin{center} {\Huge\bf On Quasifree Representations of Infinite Dimensional Symplectic Group} \\ \bigskip\bigskip \bigskip\bigskip \bigskip\bigskip {\Large Taku Matsui and Yoshihito Shimada} \\ \bigskip {\it Graduate School of Mathematics \\ Kyushu University \\ 1-10-6 Hakozaki, Fukuoka 812-8581 \\ JAPAN} \end{center} %This is also for double spacing %\newpage \vfil \noindent %This is for the abstract We consider an infinite dimensional generalization of Metaplectic representations (Weil representations) for the (double covering of) symplectic group. Given quasifree states of an infinite dimensional CCR algebra, projective unitary representations of the infinite dimensional symplectic group are constructed via unitary implementors of Bogoliubov automorphisms. Complete classification of these representations up to quasi-equivalence is obtained. \noindent \bigskip \hrule \bigskip \noindent {\bf KEY WORDS:} infinite dimensional symplectic groups, CCR algebra, quasifree state \\ \noindent {\small\tt e-mail: matsui@math.kyushu-u.ac.jp, shimada@math.kyushu-u.ac.jp} \vfil}\newpage \section{Introduction} In this paper, we consider unitary representations of an infinite symplectic group. The group $Sp(\infty)$ we deal with here is the set of invertible operators $g=1+A$ where $g$ preserves a symplectic form on an infinite dimensional vector space and $A$ is of finite rank. This group is essentially same as the inductive limit of the classical symplectic group in the sense that the latter group is dense in operator norm topology. \par First, we present a class of unitary representations of the Lie algebra $sp(\infty)$ on GNS spaces of quasifree states of the CCR (canonical commutation relations) algebra (= the infinite dimensional Heisenberg algebra). The construction of the representation is done in the same fashion as the metaplectic representation (Weil representation) of finite dimensional groups. (c.f. \cite{Lion-Vergne}) The representation constructed here is refered to as quasifree represetation. As in the finite dimensional case, our infinitesimal representations give rise to unitary representations of the double covering of $Sp(\infty)$. The infinite dimensional CCR algebra has infinitely many mutually non-equivalent representations, and we obtain a huge number of metaplectic representations of $Sp(\infty)$. This class of representations contains uncountably many irreducible representations as well as non type I factor representations. \bigskip \noindent The theory of unitary representation for infinite dimensional groups is a field of interplay between Ergodic Theory, the measure theory of infinite dimensional space, operators algebras, and mathematical physics, in particular, quantum field theory. So far two classes of infinite dimensional groups are considered. (1) groups whose matrix elements are functions: Examples are loop groups and the diffeomorphism group of the circle (See \cite{Segal}, \cite{Carey} and the references therein), and their higher dimensional analogue. (2) inductive limit of classical groups, $O(\infty)$ or $U(\infty)$. See \cite{Borodin-Olshanski}, \cite{Pickrell} and \cite{Stratila-Voiculescu}. So far the construction of unitary representations has been carried out in two ways. One way is to construct measures on an infinite dimensional space quasi-invariant under the group in question (\cite{Olshanski_Vershik}). Another method is to use Fock spaces of the quantum field theory and unitary implementors of Bogoliubov automorphisms. (c.f. \cite{Carey} and \cite{Stratila-Voiculescu}). Turning into inductive limit $O(\infty)$ and $U(\infty)$ certain representations of these groups are closely connected with the gauge invariant part of CAR (canonical anticommutation relations) algebra. In general the inductive limit procedure of compact groups yields an inductive limit of group $C^*$-algebras which are approximately finite dimensional. Then there is a one to one correspondence of primitive ideals of the AF algebra and factor representations of the group. The factor representation of $U(\infty)$ constructed on GNS spaces of quasifree states of the CAR algebra corresponds to the U(1) gauge invariant part of the CAR algebra as the quotient by the primitive ideal. (c.f. \cite{Stratila-Voiculescu}) In the same manner, spin representations of $O(\infty)$ corresponds to the $Z_2$ invariant of the CAR algebra. Quasi-equivalence of quasifree states for gauge invariant CAR algebras was investigated in \cite{Stratila-Voiculescu}, \cite{Matsui1} and \cite{Matsui} and these results leads to classification of representations of $O(\infty)$ and $U(\infty)$ on GNS spaces of quasifree states on CAR algebras. In the same spirit, we can introduce quasifree representation (= Metaplectic or Weil representation) for $Sp(\infty)$ with the aid of quasifree states of the CCR algebra. However, there is a crucial difference. The symplectic group $Sp(N)$ is non compact on one hand, and the CCR algebra is an unbounded operator algebra. The unitary representative of $O(\infty)$ or $U(\infty)$ is an element of the gauge invariant CAR algebra while this is not the case for $Sp(\infty)$. In this sense, it is not correct that classification Metaplectic representations reduces to the representation theory of the $Z_2$ gauge invariant part of the CCR algebra. Nevertheless we succeeded complete classification of generalized Metaplectic representations on GNS spaces associated with quasifree states of the CCR algebra. The main result of this paper is Theorem 7.1 where we obtained the complete classification of quasifree representation constructed in the GNS representations associated with quasifree states of the CCR algebra. To achieve our object we found it necessary to use Modular theory of von Neumann algebra for quasifree states of CCR algebras. This machinery was established by H.Araki in \cite{Araki-CCR-1}, \cite{Araki-CCR-2}, \cite{Araki-Fock-rep-CAR} and \cite{Radon-Nikodym-theorem}. Next we mention the organization of this paper. In Section 2 and 3 we introduce quasifree states of CCR algebras and Fock spaces in an abstract way. The infinitesimal quasifree representation of the Lie algebra $sp(\infty)$ is defined in Section4. If the quasifree state of the CCR algebra is pure, the associated representation of $sp(\infty)$ decomposes into two mutually non-equivalent irreducible representations. This fact is proved in Section 5. Section 6 is devoted to an analysis of von Neumann algebras generated by $sp(\infty)$. Using results of Section 6 our main result Theorem 7.1 is proved in Section 7. In the final section we show that our irreducible quasifree representation is extendible to a projective unitary representation of a larger symplectic group $Sp(P,\infty)$ where $Sp(P,\infty)$ is a symplectic transformation commuting with a fixed projection $P$ modulo Hilbert Schimidt class operators. This result is closely connected with another result of D.Pickrell in \cite{Pickrell} where he introduced the notion of spherical representations and examined the same extension property. \section{Quasifree Representations of CCR algebra} We briefly sketch GNS representations of CCR algebra associated to quasifree states. \begin{defini} \textit{Let} \emph{\( K \)} \textit{be a complex vector space and \( \gamma (f,g) \) be a non-degenerate hermitian form for \( f,g\in K \). Let \( \Gamma \) be an antilinear involution satisfying \( \Gamma ^{2}=1,\gamma (\Gamma f,\Gamma g)=-\gamma (g,f) \). A self-dual CCR algebra} \textit{\emph{\( \mathfrak {A}(K,\gamma , \Gamma ) \)}} \textit{is a complex} \textit{\emph{{*}-}}\textit{algebra generated by identity \( 1 \) and \( \{B(f)\, |\, f\in K\} \) where \( B(f) \) is complex linear in \( f\in K \) and satisfies \( B(f)^{*}=B(\Gamma f) \), \( B(f)^{*}B(g)-B(g)B(f)^{*}=\gamma (f,g) \).}\end{defini} \begin{defini} \textit{A state \( \varphi \) on} \textit{\emph{\( \mathfrak {A}(K,\gamma ,\Gamma ) \)}} \textit{is a called quasifree state} \textit{\emph{}}\textit{if \begin{gather*} \varphi (B(f_{1})\ldots B(f_{2n-1}))=0,\\ \varphi (B(f_{1})\ldots B(f_{2n}))=\sum _{\sigma \in \mathfrak {S}}\prod ^{n}_{j=1}\varphi (B(f_{\sigma (j)})B(f_{\sigma (n+j)})). \end{gather*} \( \mathfrak {S} \) is the set of all permutations of \( \{1,2,\ldots ,n\} \) satisfying} \[ \sigma (1)<\sigma (2)<\ldots <\sigma (n),\, \sigma (j)<\sigma (j+n),j=1,2,\ldots ,n.\] The cardinal number of \( \mathfrak {S} \) equals \( (2n)!2^{-n}(n!)^{-1} \). \end{defini} For any quasifree state \( \varphi \), let \[ S(f,g):=\varphi (B(f)^{*}B(g)).\] Then a positive semi-definite hermitian form \( S(\cdot ,\cdot ) \) satisfies \begin{equation} \label{eq:S(f,g)} S(f,g)-S(\Gamma g,\Gamma f)=\gamma (f,g). \end{equation} Conversely, a positive semi-definite hermitian form \( S:K\times K\rightarrow \mathbf{C} \) satisfying \eqref{eq:S(f,g)} is given. Then there exists an unique quasifree state \( \varphi _{S} \) on \emph{\( \mathfrak {A}(K,\gamma ,\Gamma ) \)} such that \[ \varphi _{S}(B(f)^{*}B(g))=S(f,g)\] for all \( f,g\in K \). That is to say, a quasifree state is completely specified by a positive semi-definite hermitian form satisfying \eqref{eq:S(f,g)}. (See \cite{Araki-CCR-1}.) We define a bounded operator ``\( S \)'' induced by a positive semi-definite hermitian form \( S \) satisfying \eqref{eq:S(f,g)}. Due to the non-degeneracy of \( \gamma \), a hermitian form \[ (f,g)_{S}:=S(f,g)+S(\Gamma g,\Gamma f)\] is positive definite. In other words \( (\cdot ,\cdot )_{S} \) is an inner product. Let \( K_{S} \) be the completion of \( K \) with respect to \( (\cdot ,\cdot )_{S} \). Then there exists a bounded operator \( S \) on \( K_{S} \) such that \[ S(f,g)=(f,Sg)_{S}\] for all \( f,g\in K \). Let \( \Gamma _{S} \) be an antiunitary involution on \( K_{S} \) such that \( \Gamma _{S}f=\Gamma f \) for all \( f\in K \). The bounded operator \( S \) satisfies \( S^{*}=S \), \( \Gamma _{S}S\Gamma _{S}=1-S \) and \( 0\leq S\leq 1 \). \( \gamma _{S}:=2S-1 \) satisfies \( \gamma (f,g)=(f,\gamma _{S}g)_{S} \) for all \( f,g\in K \). If the bounded operator \( S \) on \( K_{S} \) induced by a positive semi-definite hermitian form satisfying \eqref{eq:S(f,g)} is a projection, we call \( S \) a \emph{basis projection}. Let \( (\mathcal{H}_{S},\pi _{S},\Omega _{S}) \) be a GNS representation of CCR algebra \( \mathfrak {A}(K,\gamma ,\Gamma ) \) associated to \( \varphi _{S} \). The Hilbert space given by GNS construction is abstract, however in case that \( S \) is a basis projection, it can be written concretely. Let \( L \) be a Hilbert space and consider the Boson Fock space : \begin{gather*} \mathcal{F}_{\mathrm{b}}(L):=\bigoplus ^{\infty }_{n=0}\otimes ^{n}_{\mathrm{s}}L,\quad \otimes ^{0}_{\mathrm{s}}L:=\mathbf{C},\quad \Psi =1,\\ \left\langle f_{1}\otimes _{\mathrm{s}}\ldots \otimes _{\mathrm{s}}f_{n},g_{1}\otimes _{\mathrm{s}}\ldots \otimes _{\mathrm{s}}g_{m}\right\rangle =\delta _{mn}\frac{1}{n!}\sum _{\sigma \in \mathfrak {S}_{n}}\prod _{j=1}^{n}(f_{\sigma (j)},g_{\sigma (j)}). \end{gather*} \( \otimes _{\mathrm{s}} \) is the symmetric tensor product and \( \mathfrak {S}_{n} \) is the set of all permutations of \( \{1,2,\ldots ,n\} \). Now we define annihilation operators \( b(f),f\in L \) on \( \mathcal{F}_{\mathrm{b}}(L) \) as follows : \begin{gather*} b(f)f_{1}\otimes _{\mathrm{s}}\ldots \otimes _{\mathrm{s}}f_{n}:=\frac{1}{\sqrt{n}}\sum _{j=1}^{n}(f,f_{j})f_{1}\otimes _{\mathrm{s}}\ldots \otimes _{\mathrm{s}}f_{j-1}\otimes _{\mathrm{s}}f_{j+1}\ldots \otimes _{\mathrm{s}}f_{n},\\ b(f)\Psi :=0 \end{gather*} and creation operators \( b^{\dagger }(f),f\in L \) on \( \mathcal{F}_{\mathrm{b}}(L) \) as follows : \emph{ \begin{gather*} b^{\dagger }(f_{1})f_{2}\otimes _{\mathrm{s}}\ldots \otimes _{\mathrm{s}}f_{n+1}:=\sqrt{n+1}f_{1}\otimes _{\mathrm{s}}\ldots \otimes _{\mathrm{s}}f_{n+1},\\ b^{\dagger }(f)\Psi :=f. \end{gather*} } \begin{lm}\label{thm:CCR-Fock-rep} \emph{Suppose that} \( S:K_{S}\rightarrow K_{S} \) \emph{is a basis projection}. \begin{enumerate} \item \emph{\( b^{\dagger }(f),b(g),f,g\in SK_{S} \) are closable operators}. \emph{Let \( \overline{A} \) be the closure of operator \( A \)}. \emph{The finite particle vector subspace of \( \mathcal{F}_{\mathrm{b}}(SK_{S}) \) is a core for all \( \overline{b^{\dagger }(f)},\overline{b(g)},f,g\in SK_{S} \)}. \item \emph{Due to} (1), \emph{we can define the addition and multiplication of creation and annihilation operators on the finite particle vector subspace of \( \mathcal{F}_{\mathrm{b}}(SK_{S}) \). Let \( \mathfrak {A}_{\mathbf{CCR}}(SK_{S}) \) be a} {*}-\emph{algebra} \emph{generated by all annihilation and creation operators}. \emph{Let} \emph{\( \alpha (S):\mathfrak {A}(K,\gamma ,\Gamma )\rightarrow \mathfrak {A}_{\mathbf{CCR}}(SK_{S}) \) be a} {*}-\emph{homomorphism} \emph{satisfying the following relation} : \emph{ \[ \alpha (S)(B(f)):=b^{\dagger }(Sf)+b(S\Gamma f),f\in K.\] Then \( (\mathcal{F}_{\mathrm{b}}(SK_{S}),\alpha (S),\Psi ) \) is a} {*}-\emph{representation of CCR algebra \( \mathfrak {A}(K,\gamma ,\Gamma ) \). Moreover}, \emph{it is unitary equivalent to the GNS representation} \( (\mathcal{H}_{S},\pi _{S},\Omega _{S}). \) \end{enumerate} \end{lm} \begin{proof} (1) See chapter X section 7 of \cite{Reed-Simon-2}. (2) An unitary operator \( u:\mathcal{H}_{S}\rightarrow \mathcal{F}_{\mathrm{b}}(SK_{S}), \) \( \pi _{S}(X)\Omega _{S}\mapsto \alpha (S)(X)\Psi \) satisfies \( u^{*}\alpha (S)(X)u=\pi _{S}(X) \) for all \( X\in \mathfrak {A}(K,\gamma ,\Gamma ) \).\end{proof} By Lemma \ref{thm:CCR-Fock-rep}, we call \( \pi _{S} \) a \emph{Fock representation} and \( \varphi _{S} \) a \emph{Fock state} if \( S \) is a basis projection. In case that \( \varphi _{S} \) is a Fock state, we have the following important lemma. (See Lemma 5.4 and 5.5 of \cite{Araki-CCR-1}.) \begin{lm}\label{thm:Fock-rep-property} \emph{Suppose that} \( S:K_{S}\rightarrow K_{S} \) \emph{is a basis projection}. \emph{} \begin{enumerate} \item \( \pi _{S}(B(f)),f\in \mathrm{Re}K \) \emph{is an essentially self-adjoint operator and set} \[ W_{S}(f):=\exp \left( i\overline{\pi _{S}(B(f)}\right) .\] \emph{Then \( W_{S}(f) \) satisfies the following relations} : \[ W_{S}(f_{1})W_{S}(f_{2})=\exp \left( -\frac{1}{2}\gamma (f_{1},f_{2})\right) W_{S}(f_{1}+f_{2}).\] \item \emph{If \( f\in \mathrm{Re}K_{S} \)}, \emph{we define \( W_{S}(f) \) via the following limit \begin{equation} \label{eq:limW(f_n)} W_{S}(f):=\mathrm{s}\frac{\, \, }{\, \, }\lim _{n\rightarrow \infty }W_{S}(f_{n}) \end{equation} where} \( \{f_{n}\} \) \emph{is a sequence in} \( \mathrm{Re}K \) \emph{satisfying} \( \left\Vert f-f_{n}\right\Vert \rightarrow 0 \). \emph{Note that the limit} \eqref{eq:limW(f_n)} \emph{does not depend on the choice of} \( \{f_{n}\} \). \item \emph{Let} \( \mathrm{Re}K_{S}:=\{f\in K_{S}\, |\, \Gamma _{S}f=f\} \)\emph{. The restriction of} \( (\cdot ,\cdot )_{S} \) \emph{to \( \mathrm{Re}K_{S} \) is an inner product of \( \mathrm{Re}K_{S} \).} \( f\mapsto W_{S}(f) \) \emph{is continuous with respect to the norm on} \( \mathrm{Re}K_{S} \) \emph{and the strong operator topology of bounded operators on} \( \mathcal{H}_{S} \). \item \emph{Let \( L \) be a subspace of \( \mathrm{Re}K_{S} \). Let \( L^{\vee } \) be the set of vectors \( f\in \mathrm{Re}K_{S} \) such that \( (f,\gamma _{S}g)_{S}=0 \) for all \( g\in L \) and let \( \overline{L} \) be the closure of \( L \) in} \( \mathrm{Re}K_{S} \). \emph{Let \( \mathcal{R}_{S}(L) \) be a von Neumann algebra generated by} \( W_{S}(f),f\in L. \) \emph{Then we obtain the following relations} : \begin{enumerate} \item \( \mathcal{R}_{S}(L)=\mathcal{R}(\overline{L}), \) \item \( \mathcal{R}_{S}(L)'=\mathcal{R}_{S}(L^{\vee }), \) \item \( \{\mathcal{R}_{S}(L_{1})\cup \mathcal{R}_{S}(L_{2})\}''=\mathcal{R}_{S}(L_{1}+L_{2}), \) \item \( \mathcal{R}_{S}(L_{1})\cap \mathcal{R}_{S}(L_{2})=\mathcal{R}_{S}(\overline{L_{1}}\cap \overline{L_{2}}). \) \end{enumerate} \end{enumerate} \end{lm} We introduce an another hermitian form \( \widehat{\gamma }_{S} \) on \( K_{S}\oplus K_{S} \) via the following relation : \[ \widehat{\gamma }_{S}(f_{1}\oplus g_{1},f_{2}\oplus g_{2}):=(f_{1},\gamma _{S}f_{2})_{S}-(g_{1},\gamma _{S}g_{2})_{S}\] for all \( f_{i},g_{i}\in K_{S} \). Set \( \widehat{\Gamma }_{S}:=\Gamma _{S}\oplus \Gamma _{S} \). Then \( \widehat{\gamma }_{S} \) satisfies \( \hat{\gamma }_{S}(\hat{\Gamma }_{S}h_{1},\hat{\Gamma }_{S}h_{2})=-\hat{\gamma }_{S}(h_{1},h_{2}) \) for all \( h_{i}\in K_{S}\oplus K_{S} \). \[ P_{S}(f_{1}\oplus f_{2},g_{1}\oplus g_{2}):=(f_{1},Sf_{2}+\sqrt{S(1-S)}g_{2})_{S}+(g_{1},\sqrt{S(1-S)}f_{2}+(1-S)g_{2})_{S}\] is a positive semi-definite hermitian form on \( K_{S}\oplus K_{S} \) satisfying \[ P_{S}(h_{1},h_{2})-P_{S}(\widehat{\Gamma }_{S}h_{2},\hat{\Gamma }_{S}h_{1})=\hat{\gamma }_{S}(h_{1},h_{2}).\] We denote the completion of \( K_{S}\oplus K_{S} \) with respect to the inner product \( (h_{1},h_{2})_{P_{S}}:=P_{S}(h_{1},h_{2})+P_{S}(\hat{\Gamma }_{S}h_{2},\hat{\Gamma }_{S}h_{1}) \) by \( K_{P_{S}} \). \begin{lm}\label{thm:P_S-is-a-projection} \emph{The bounded operator} \emph{\( P_{S} \) on} \( K_{P_{S}} \) \emph{satisfying} \( P_{S}(h_{1},h_{2})=(h_{1},P_{S}h_{2})_{P_{S}} \) \emph{for all} \( h_{i}\in K_{S}\oplus K_{S} \) \emph{is a basis projection}. \end{lm} \begin{proof} Let \( D(\gamma ^{-1}_{S}) \) be a domain of \( \gamma ^{-1}_{S} \). By the non-degeneracy of \( \gamma \), \( D(\gamma ^{-1}_{S}) \) is a dense set of \( K_{S} \). Let \( \widehat{\gamma }_{S}(f_{1}\oplus g_{1},f_{2}\oplus g_{2})=(f_{1}\oplus g_{1},h\oplus k)_{P_{S}} \). Then \( \gamma _{S}f_{2}=h+2\sqrt{S(1-S)}k \), \( -\gamma _{S}g_{2}=2\sqrt{S(1-S)}h+k \). If \( f_{2},g_{2}\in D(\gamma ^{-1}_{S}) \), we have \[ \gamma _{P_{S}}(f_{2}\oplus g_{2})=h\oplus k=\gamma _{S}^{-1}(f_{2}+2\sqrt{S(1-S)}g_{2})\oplus -\gamma _{S}^{-1}(2\sqrt{S(1-S)}f_{2}+g_{2}).\] By \( P_{S}=\frac{1}{2}(\gamma _{P_{S}}+1) \), \( P_{S} \) is written explicitly on \( D(\gamma ^{-1}_{S})\oplus D(\gamma ^{-1}_{S}) \) as follows : \[ P_{S}(f\oplus g)=\gamma ^{-1}_{S}(Sf+\sqrt{S(1-S)}g)\oplus -\gamma ^{-1}_{S}(\sqrt{S(1-S)}f+(1-S)g).\] It is easily checked that \( P_{S} \) is a projection on \( K_{P_{S}} \).\end{proof} \emph{Remark}. Let \( L \) be a dense set of \( K_{S} \) with respect to \( (\cdot ,\cdot )_{S} \), then \( L\oplus L \) is a dense set of \( K_{S}\oplus K_{S} \) with respect to \( (\cdot ,\cdot )_{P_{S}} \). Indeed, for any \( f\oplus g\in K_{S}\oplus K_{S} \), there exist \( f_{n},g_{n}\in L \) such that \( \parallel f_{n}-f\parallel _{S},\parallel g_{n}-g\parallel _{S}\rightarrow 0(n\rightarrow \infty ) \). By the following equation \begin{equation} \label{eq:relation_between_<,>(S)_and_<,>(P_S)} \parallel f\oplus g\parallel ^{2}_{P_{S}}=\parallel \sqrt{S}f+\sqrt{1-S}g\parallel ^{2}_{S}+\parallel \sqrt{1-S}f+\sqrt{S}g\parallel ^{2}_{S}, \end{equation} we have \( \parallel (f_{n}\oplus g_{n})-(f\oplus g)\parallel _{P_{S}}\rightarrow 0(n\rightarrow \infty ) \). \medskip By Lemma \ref{thm:P_S-is-a-projection}, \( \varphi _{P_{S}} \) is a Fock state on CCR algebra \( \mathfrak {A}(K_{S}\oplus K_{S},\widehat{\gamma }_{S},\widehat{\Gamma }_{S}). \) We denote a GNS representation of CCR algebra \( \mathfrak {A}(K_{S}\oplus K_{S},\hat{\gamma }_{S},\hat{\Gamma }_{S}) \) associated to \( \varphi _{P_{S}} \) by \( (\mathcal{H}_{P_{S}},\pi _{P_{S}},\Omega _{P_{S}}) \) . The following corollary is a consequence of the direct application of lemma \ref{thm:Fock-rep-property} to Fock representation \( (\mathcal{H}_{P_{S}},\pi _{P_{S}},\Omega _{P_{S}}) \). \begin{cor}\label{thm:factoriarity_of_R(ReK+0)} \( \mathcal{R}_{P_{S}}(\mathrm{Re}K_{S}\oplus 0)'=\mathcal{R}_{P_{S}}(0\oplus \mathrm{Re}K_{S}) \) \emph{and} \( \mathcal{R}_{P_{S}}(\mathrm{Re}K_{S}\oplus 0) \) \emph{is a factor}.\end{cor} \begin{remark}\label{thm:Remark}Let \( \alpha :\mathfrak {A}(K,\gamma ,\Gamma )\rightarrow \mathfrak {A}(K_{S}\oplus K_{S},\widehat{\gamma }_{S},\widehat{\Gamma }_{S}) \) be a {*}-homomorphism defined by \( \alpha (B(f))=B(f\oplus 0) \) and \( u_{\alpha }:\mathcal{H}_{S}\rightarrow \mathcal{H}_{P_{S}} \) be a linear operator defined by \( u_{\alpha }(\pi _{S}(A)\Omega _{S})=\pi _{P_{S}}(\alpha (A))\Omega _{P_{S}} \) for all \( A\in \mathfrak {A}(K,\gamma ,\Gamma ). \) Then \( u_{\alpha } \) preserves the inner product. In fact, since \( \varphi _{P_{S}} \) and \( \varphi _{S} \) are quasifree states and \[ \varphi _{P_{S}}(B(f\oplus 0)^{*}B(g\oplus 0))=P_{S}(f\oplus 0,g\oplus 0)=S(f,g)=\varphi _{S}(B(f)^{*}B(g))\] for all \( f,g\in K \), we have \( \varphi _{P_{S}}(\alpha (A))=\varphi _{S}(A) \) for all \( A\in \mathfrak {A}(K,\gamma ,\Gamma ). \) If \( X\) and \( Y\) are elements of \( \mathfrak {A}(K,\gamma ,\Gamma ) \) and set \( A:=X^{*}Y \), then \[ \left\langle u_{\alpha }\pi _{S}(X)\Omega _{S},u_{\alpha }\pi _{S}(Y)\Omega _{S}\right\rangle =\varphi _{P_{S}}(\alpha (X^{*}Y))=\varphi _{S}(X^{*}Y)=\left\langle \pi _{S}(X)\Omega _{S},\pi _{S}(Y)\Omega _{S}\right\rangle .\] If we identify \( u_{\alpha }\mathcal{H}_{S} \) with \( \mathcal{H}_{S} \), \( \mathcal{H}_{S} \) is a closed subspace of \( \mathcal{H}_{P_{S}} \) and \( \mathcal{H}_{S}=\mathcal{F}_{\mathrm{b}}(P_{S}(K_{S}\oplus 0)) \). Moreover, since \( u_{\alpha }\pi _{S}(A)=\pi _{P_{S}}(\alpha (A))u_{\alpha } \) on \( D(\pi _{S}):=\pi _{S}(\mathfrak {A}(K,\gamma ,\Gamma ))\Omega _{S} \), we can identify \( \pi _{S}(A) \) with \( \pi _{P_{S}}(\alpha (A))|D(\pi _{S}) \) for all \( A\in \mathfrak {A}(K,\gamma ,\Gamma ). \)\end{remark} \section{Fock Space and Exponential Vectors} In Remark \ref{thm:Remark}, \( \mathcal{H}_{S} \) is regarded as a closed subspace of \( \mathcal{H}_{P_{S}} \). We explain the point in detail. Let \( L_{1},L_{2} \) be Hilbert spaces and \( L:=L_{1}\oplus L_{2} \) and \( e(u):=\sum _{n=0}^{\infty }(\sqrt{n!})^{-1}\otimes _{\mathrm{s}}^{n}u \) for all \( u\in L \). We call \( e(u) \) an \emph{exponential vector}. \begin{lm}\label{thm:decomposition_of_Fock_space} If \( u_{1}\in L_{1} \) and \( u_{2}\in L_{2} \), then there exists an unique unitary operator \( U:\mathcal{F}_{\mathrm{b}}(L)\rightarrow \mathcal{F}_{\mathrm{b}}(L_{1})\otimes \mathcal{F}_{\mathrm{b}}(L_{2}) \) such that \( Ue(u_{1}+u_{2})=e(u_{1})\otimes e(u_{2}) \). This shows \[ \mathcal{F}_{\mathrm{b}}(L)=\mathcal{F}_{\mathrm{b}}(L_{1})\otimes \mathcal{F}_{\mathrm{b}}(L_{2}).\] (See chapter II section 19 of \cite{K.R.Parthasarathy}.)\end{lm} \begin{lm}\label{thm:decomposition_of_PK} \( P_{S}K_{P_{S}}=[P_{S}(K_{S}\oplus 0)]\oplus [0\oplus E_{S}(\{0\})K_{S}]=[P_{S}(0\oplus K_{S})]\oplus [E_{S}(\{1\})K_{S}\oplus 0] \) \emph{where} \( E_{S}(B) \) \emph{is the spectral projection of \( S \) for a Borel set} \( B\subset \mathbf{R} \). \end{lm} \begin{proof} The restriction of \( (\cdot ,\cdot )_{P_{S}} \) to \( P_{S}K_{P_{S}} \) is an inner product of \( P_{S}K_{P_{S}} \). Let \( [P_{S}(K_{S}\oplus 0)]^{\perp } \) be the orthogonal complement of \( P_{S}(K_{S}\oplus 0) \) in \( P_{S}K_{S} \). Let \( u\oplus v\in [P_{S}(K_{S}\oplus 0)]^{\perp } \), then we have \[ 0=(u\oplus v,P_{S}(f\oplus 0))_{P_{S}}=(Su+\sqrt{S(1-S)}v,f)_{S}\] for all \( f\in K_{S} \). This implies \( \sqrt{S}(\sqrt{S}u+\sqrt{1-S}v)=0 \), i.e. \( Su+\sqrt{1-S}v\in E_{S}(\{0\})K_{S} \). By \( P_{S}(u\oplus v)=u\oplus v \) and \( \gamma ^{-1}_{S}=-1,\sqrt{1-S}=1,\sqrt{S}=0 \) on \( E_{S}(\{0\})K_{S} \), we have \[ u\oplus v=P(u\oplus v)=0\oplus (\sqrt{S}u+\sqrt{1-S}v).\] Thus \( u=0 \), \( v\in E_{S}(\{0\})K_{S} \). We obtain \( [P_{S}(K_{S}\oplus 0)]^{\perp }\subset 0\oplus E_{S}(\{0\})K_{S}. \) Converse relation is seen from direct computation. \end{proof} From Lemma \ref{thm:decomposition_of_Fock_space}, Lemma \ref{thm:decomposition_of_PK} and Remark \ref{thm:Remark}, we obtain the factorization of the Fock space : \begin{equation} \label{eq:H(P_S)_equal_H(S)xL(S)} \mathcal{H}_{P_{S}}=\mathcal{H}_{S}\otimes \mathcal{L}_{S}, \end{equation} \( \mathcal{L}_{S}:=\mathcal{F}_{\mathrm{b}}(0\oplus E_{S}(\{0\})K_{S}) \). In particular, if \( 0q_S=q_P(S)} \( q_{S}^{\sigma } \) \emph{and} \( q_{P_{S}}^{\sigma } \) \emph{are quasi-equivalent}.\end{cor} \begin{proof} \( \Omega _{P_{S}} \) is a separating vector for \( C(\mathcal{M}_{P_{S}}^{+}) \). Indeed, let \[ \left( \begin{array}{cc} a_{+}\otimes 1_{+} & 0\\ 0 & a_{-}\otimes 1_{-} \end{array}\right) \Omega _{P_{S}}=0,\, a_{\sigma }\in C(\mathcal{M}_{S}^{\sigma }),\] then \( a_{+}\Omega _{S}=0 \). Since \( \Omega _{S} \) is a separating vector for \( C(\mathcal{M}_{S}^{+}) \), we have \( a_{+}=0 \). Moreover, by \( q_{S}^{+}\sim _{q}q_{S}^{-} \)(Lemma \ref{thm:S_not-projection=q_S==q_S(+)==q_S(-)}), we have \( a_{-}=0 \) as well. Let \( \widehat{E}_{+}:\mathcal{H}^{+}_{P_{S}}\rightarrow \mathcal{H}_{S}^{+} \) be a projection and \( C(\widehat{E}_{+}):=\min \{E\in C(\mathcal{M}_{P_{S}}^{+})\, |\, E^{2}=E^{*}=E,E\geq \widehat{E}_{+}\} \). Since \( C(\widehat{E}_{+})\Omega _{P_{S}}=\Omega _{P_{S}} \) and \( \Omega _{P_{S}} \) is a separating vector for \( C(\mathcal{M}_{P_{S}}^{+}) \), we have \( C(\widehat{E}_{+})=1 \), that is, \( q_{S}^{+}\sim _{q}q_{P_{S}}^{+} \). By Lemma \ref{thm:S_not-projection=q_S==q_S(+)==q_S(-)}, we obtain \[q_{S}^{+}\sim _{q}q_{P_{S}}^{+}\sim _{q}q_{S}^{-}\sim _{q}q_{P_{S}}^{-}.\] \end{proof} If \( 0\alpha >0 \) such that \( \alpha \parallel f\parallel _{S}\leq \parallel f\parallel _{S'}\leq \beta \parallel f\parallel _{S} \) for all \( f\in K \).} \item \textit{\( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) is a Hilbert-Schmidt class operator on \( K_{S} \)} \textit{\emph{}}\textit{where \( \chi (S):=\tanh ^{-1}2\sqrt{S(1-S)} \) and \( \rho (S):=(2S-1)^{-1}|2S-1| \).} \end{enumerate} \end{thm} By the equivalence of norms \( \parallel \cdot \parallel _{S} \) and \( \parallel \cdot \parallel _{S'} \), we can see a bounded operator \( S' \) on \( K_{S'} \) as a bounded operator on \( K_{S} \). \begin{lm} \emph{If} \( \alpha \parallel \cdot \parallel _{S}\leq \parallel \cdot \parallel _{S'}\leq \beta \parallel \cdot \parallel _{S} \), \emph{there exists} \( 0<\alpha '<\beta ' \) \emph{such that} \( \alpha '\parallel \cdot \parallel _{P_{S}}\leq \parallel \cdot \parallel _{P_{S'}}\leq \beta '\parallel \cdot \parallel _{P_{S}} \).\end{lm} \begin{proof} Immediate from \eqref{eq:relation_between_<,>(S)_and_<,>(P_S)}.\end{proof} \begin{lm}\label{thm:P(S)-P(S'):H.S.<==>1-exp(-x(S))exp(x(S')):H.S.} \emph{Let} \( S,\, S'\in \mathfrak {S} \) \emph{and} \textit{the topologies induced by \( \parallel f\parallel _{S} \) and \( \parallel f\parallel _{S'} \) on \( K \) are equivalent}. \emph{Then the following conditions are equivalent}. \begin{enumerate} \item \( P_{S}-P_{S'} \) \emph{is a Hilbert-Schimdt class operator}, \item \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) \emph{is a Hilbert-Schmidt class operator}, \item \( 1-\rho (S')e^{-\chi (S')}e^{\chi (S)}\rho (S) \) \emph{is a Hilbert-Schmidt class operator}. \end{enumerate} \end{lm} \begin{proof} See Lemma 6.5 of \cite{Araki-CCR-2}.\end{proof} \begin{lm}\label{thm:Unitary_Induced_by_Bogoliubov_trans} \emph{Let} \( S,S'\in \mathfrak {S} \) \emph{be basis projections and assume that} \( K=K_{S}=K_{S'} \). \begin{enumerate} \item \emph{Let} \( \theta (S,S') \) \emph{be a non-negative hermitian operator on} \( K \) \emph{satisfying} \( \sinh ^{2}\theta (S,S')=-(S-S')^{2} \). \emph{Let} \begin{gather*} u_{12}(S/S'):=(\sinh \theta (S,S')\cosh \theta (S,S'))^{-1}SS'(1-S), \\ u_{21}(S/S'):=-(\sinh \theta (S,S')\cosh \theta (S,S'))^{-1}(1-S)S'S, \\ H(S/S'):=-i\theta (S,S')\{u_{12}(S/S')+u_{21}(S/S')\}. \end{gather*} \emph{Then \( u_{ij}(S/S')^{*}=u_{ji}(S/S') \)} \emph{and} \( H(S/S') \) \emph{satisfies} \begin{gather*} H(S/S')^{\dagger }=H(S/S'),\quad \Gamma H(S/S')\Gamma =-H(S/S'), \\ (iH(S/S'))^{*}=iH(S/S'). \end{gather*} ({*} \emph{is relative to \( (\cdot ,\cdot )_{S} \)}.) \emph{Let} \[ U(S/S'):=\exp (iH(S/S')). \] \( U(S/S') \) \emph{satisfies} \begin{gather*} U(S/S')^{\dagger }U(S/S')=U(S/S')U(S/S')^{\dagger }=1,\quad [\Gamma ,U(S/S')]=0, \\ U(S/S')^{\dagger }SU(S/S')=S'. \end{gather*} \item \( S-S' \) \emph{is a Hilbert-Schmidt class operator if and only if} \( \theta (S,S') \) \emph{is a Hilbert-Schmidt class operator}. \item \emph{Let \( \theta (S,S') \) be a Hilbert-Schmidt class operator. Then there exists an unique unitary operator} \( T(S,S')\in \mathcal{M}_{S} \) \emph{such that} \[ T(S,S')^{*}\overline{\pi _{S}(A)}T(S,S')=\pi _{S}[\tau (U(S/S'))A]\] \emph{on} \( D(\pi _{S}) \) \emph{and} \begin{equation} \label{eq:} \left\langle \Omega _{S},T(S,S')\Omega _{S}\right\rangle =\mathrm{det}_{SK}\left( \frac{1}{\sqrt{\cosh \theta (S,S')}}\right) \end{equation} \emph{where} \( \mathrm{det}_{SK} \) \emph{is the determinant of} \( SK \).(\emph{Since} \( \theta (S,S') \) \emph{commutes with} \( S \), \emph{the right hand side of} \eqref{eq:} \emph{is well-defined}.) \end{enumerate} \end{lm} \begin{proof} (1) See Lemma 5.4 of \cite{Araki-CCR-2}. (3) See Lemma 5.5 of \cite{Araki-CCR-2}.\end{proof} \begin{lm}\label{thm:S-S':H.S-->q(S)_q(S'):unitary_eqiv} \emph{Assume that} \( S,\, S'\in \mathfrak {S} \) \emph{are basis projections and} \textit{the topologies induced by \( \parallel f\parallel _{S} \) and \( \parallel f\parallel _{S'} \) on \( K \) are equivalent}. \emph{If} \( S-S' \) \emph{is a Hilbert-Schmidt class operator}, \emph{then} \( q_{S}^{\sigma } \) \emph{and} \( q_{S'}^{\sigma } \) \emph{are unitary equivalent}.\end{lm} \begin{proof} Let \[ V\pi _{S'}(A)\Omega _{S'}=\overline{\pi _{S}(A)}T(S,S')\Omega _{S},\quad A\in \mathfrak {A}(K,\gamma ,\Gamma ).\] Since \( U(S/S')^{\dagger }SU(S/S')=S' \), we have \( \varphi _{S'}=\varphi _{S}\circ \tau (U(S/S')) \). This shows that \( V \) is an unitary operator from \( \mathcal{H}_{S'} \) to \( \mathcal{H}_{S} \) and satisfies \( V\pi _{S'}(A)=\overline{\pi _{S}(A)}V \) on \( D(\pi _{S'}) \) for all \( A\in \mathfrak {A}(K,\gamma ,\Gamma ) \). By \( V\mathcal{H}_{S'}^{\sigma }\subset \mathcal{H}_{S}^{\sigma } \), the restriction of \( V \) to \( \mathcal{H}_{S'}^{\sigma } \) is an unitary operator from \( \mathcal{H}_{S'}^{\sigma } \) to \( \mathcal{H}_{S}^{\sigma } \) and we have \( VQ^{\sigma }_{S'}(H)=Q_{S}^{\sigma }(H)V \) for all \( H\in sp(\infty ) \). Thus \( q_{S}^{\sigma } \) is unitary equivalent to \( q_{S'}^{\sigma } \).\end{proof} The next corollary is directly seen from the above lemma. \begin{cor}\label{thm:P(S)-S(S'):H.S.=>q_P(S)=q_P(S')} \emph{If} \textit{the topologies induced by \( \parallel f\parallel _{S} \) and \( \parallel f\parallel _{S'} \) on \( K \) are equivalent} \emph{and} \( P_{S}-P_{S'} \) \emph{is a Hilbert-Schmidt class operator}, \emph{then} \( q_{P_{S}}^{\sigma } \) \emph{and} \( q_{P_{S'}}^{\sigma } \) \emph{are unitary equivalent}.\end{cor} \begin{lm} \emph{Assume that} \( S,\, S'\in \mathfrak {S} \) \emph{are not projection}. \emph{If} \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) \emph{is a Hilbert-Schmidt class operator}, \emph{then} \( q_{S}^{\sigma } \) \emph{and} \( q_{S'}^{\sigma } \) \emph{are quasi-equivalent}. \end{lm} \begin{proof} Immediate from Corollary \ref{thm:S:non-projection==>q_S=q_P(S)}, Lemma \ref{thm:P(S)-P(S'):H.S.<==>1-exp(-x(S))exp(x(S')):H.S.} and Corollary \ref{thm:P(S)-S(S'):H.S.=>q_P(S)=q_P(S')}.\end{proof} \begin{lm} \emph{Let} \( S\in \mathfrak {S} \) \emph{be a projection and} \( S'\in \mathfrak {S} \) \emph{be not a projection}. \emph{Then} \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) \emph{is not a Hilbert-Schmidt class operator}.\end{lm} \begin{proof} Suppose \( q^{+}_{S}\sim _{q}q^{+}_{S'} \) and \( q_{S}^{-}\sim _{q}q^{-}_{S'} \). Then since \( q^{+}_{S}\sim _{q}q_{S}^{-} \) and the irreducibility of \( q_{S}^{\sigma } \), we have \( q^{+}_{S}\sim q_{S}^{-} \). However, since \( S \) is a projection, \( q_{S}^{+}\not \sim q_{S}^{-} \). This is contradiction. So \( q_{S}^{+}\not \sim _{q}q_{S'}^{+} \) or \( q_{S}^{-}\not \sim _{q}q_{S'}^{-} \). If \( q_{S}^{+}\sim _{q}q_{S'}^{+} \) and \( q_{S}^{-}\not \sim _{q}q_{S'}^{-} \), then \( P_{S}-P_{S'} \) is not a Hilbert-Schmidt class operator from \( q_{P_{S'}}^{+}\sim _{q}q_{S'}^{+}\sim _{q}q_{S}^{+}\not \sim _{q}q^{+}_{P_{S}} \). Thus \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) is not a Hilbert-Schmidt class operator. The case of \( q_{S}^{-}\sim _{q}q^{-}_{S'} \) and \( q_{S}^{+}\not \sim _{q}q_{S'}^{+} \) is quite similar. If \( q_{S}^{+}\sim _{q}q^{-}_{S'} \) and \( q_{S}^{+}\not \sim _{q}q_{S'}^{+} \) and \( q_{S}^{-}\not \sim _{q}q_{S'}^{-} \), we have \( q_{P_{S'}}^{+}\sim _{q}q^{-}_{P_{S'}}\sim _{q}q_{S'}^{-}\sim _{q}q_{S}^{+}\not \sim _{q}q_{P_{S}}^{+} \). Thus \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) is not a Hilbert-Schmidt class operator. The case of \( q_{S}^{+}\not \sim _{q}q^{-}_{S'} \) and \( q_{S}^{+}\not \sim _{q}q_{S'}^{+} \) and \( q_{S}^{-}\not \sim _{q}q_{S'}^{-} \) is trivial. Therefore, \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) is not a Hilbert-Schmidt class operator.\end{proof} From the above lemmas, we have the necessary condition of Theorem \ref{thm:Main_Theorem}. \begin{lm}[Necessity of Theorem \ref{thm:Main_Theorem}] \emph{Suppose that} \( S,\, S'\in \mathfrak {S} \) \emph{satisfy the following two conditions}. \begin{enumerate} \item \textit{The topologies induced by \( \parallel f\parallel _{S} \) and \( \parallel f\parallel _{S'} \) on \( K \) are equivalent}, \item \textit{\( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) is a Hilbert-Schmidt class operator.} \end{enumerate} \emph{Then two representations} \( q_{S}^{\sigma } \) \emph{and} \( q_{S'}^{\sigma } \) \emph{are quasi-equivalent}. \end{lm} Next, we prove the sufficiency of Theorem \ref{thm:Main_Theorem}. A state \( \varphi _{P_{S}} \) on CCR algebra \( \mathfrak {A}(K_{S}\oplus K_{S},\widehat{\gamma }_{S},\widehat{\Gamma }_{S}) \) can be viewed as a state on \( \mathcal{M}_{S}^{+} \) satisfying \( \varphi _{P_{S}}(Q):=\left\langle \Omega _{P_{S}},Q\Omega _{P_{S}}\right\rangle \), \( Q\in \mathcal{M}_{P_{S}}^{+} \). Now let \( \dim K<\infty \). Since \( P_{S}-P_{S'} \) is a Hilbert-Schmidt class operator, \( q_{P_{S}}^{+} \) and \( q_{P_{S'}}^{+} \) are unitary equivalent and we can identify \( \mathcal{M}_{P_{S}}^{+} \) with \( \mathcal{M}_{P_{S'}}^{+} \). Therefore, a state \( \varphi _{P_{S'}} \) on CCR algebra \( \mathfrak {A}(K_{S'}\oplus K_{S'},\widehat{\gamma }_{S'},\widehat{\Gamma }_{S'}) \) is regarded as a state on \( \mathcal{M}^{+}_{P_{S}} \) satisfying \( \varphi _{P_{S'}}(Q):=\left\langle \Omega ',Q\Omega '\right\rangle \), \( \Omega ':=T(P_{S},P_{S'})\Omega _{P_{S}} \), \( Q\in \mathcal{M}^{+}_{P_{S'}}=\mathcal{M}^{+}_{P_{S}} \) where \( T(P_{S},P_{S'}) \) is an unitary operator determined by (3) of Lemma \ref{thm:Unitary_Induced_by_Bogoliubov_trans}. We quote the following two results to prove the sufficiency of the main theorem. \begin{lm}\label{thm:THEOREM4(5)} \emph{Let} \( \mathcal{M} \) \emph{be a von Neumann algebra on a Hilbert space \( \mathcal{H} \) with a cyclic and separating vector} \( \Psi \). \emph{Let} \( V_{\Psi } \) \emph{be a natural positive corn associated with the pair} \( (\mathcal{M},\Psi ) \). \emph{If} \( \Phi \) \emph{is an another cyclic and separating vector for} \( \mathcal{M} \), \emph{then} \( \Phi \in V_{\Psi } \) \emph{if and only if the following 2 conditions hold} : \begin{enumerate} \item \( J_{\Phi }=J_{\Psi } \), \item \( \left\langle \Phi ,Q_{+}\Psi \right\rangle \geq 0 \) \emph{for all} \( Q_{+}\in \mathcal{M}\cap \mathcal{M}',\, Q_{+}\geq 0 \). \end{enumerate} \end{lm} \begin{proof} See THEOREM 4 (5) of \cite{Radon-Nikodym-theorem}.\end{proof} \begin{lm}\label{thm:THEOREM4(8)} \emph{Let} \( \mathcal{M} \) \emph{be a von Neumann algebra on a Hilbert space \( \mathcal{H} \) with a cyclic and separating vector} \( \Psi \) \emph{and let \( V_{\Psi } \) be the natural positive corn for \( \Psi \). Let \( \Phi _{i}\in V_{\Psi }(i=1,2) \) and \( \varphi _{\Phi _{i}} \) be a vector state for \( \Phi _{i} \). Then} \[ \parallel \varphi _{\Phi _{1}}-\varphi _{\Phi _{2}}\parallel \geq \parallel \Phi _{1}-\Phi _{2}\parallel ^{2}.\] \end{lm} \begin{proof} See THEOREM 4 (8) of \cite{Radon-Nikodym-theorem}.\end{proof} \begin{lm} \emph{Let \( \dim K<\infty \) and \( S,\, S'\in \mathfrak {S} \) be} \( 0||w-w'||^2} \parallel (\varphi _{P_{S}}-\varphi _{P_{S'}})|\mathcal{M}^{+}_{P_{S}}\parallel \geq 2\left\{ 1-\mathrm{det}_{P_{S}K_{P_{S}}}\left( \frac{1}{^{4}\sqrt{P_{S}P_{S'}P_{S}}}\right) \right\} \end{equation} \end{lm} \begin{proof} Let \( J_{\Omega '} \) and \( \Delta _{\Omega '} \) be the modular conjugation and modular operator associated with the pair \( (\mathcal{M}^{+}_{P_{S}},\Omega ') \) and let \( V_{\Omega _{P_{S}}} \) be the natural positive corn associated with the pair \( (\mathcal{M}_{P_{S}}^{+},\Omega _{P_{S}}) \). We show \( J_{\Omega '}=J_{\Omega _{P_{S}}} \) and \( \left\langle \Omega ',Q\Omega _{P_{S}}\right\rangle \geq 0 \) for all \( Q\in C(\mathcal{M}_{P_{S}}^{+}),\, Q\geq 0 \) with help of Lemma \ref{thm:THEOREM4(5)} to prove \( \Omega '\in V_{\Omega _{P_{S}}} \) . We prove the first part, \( J_{\Omega '}=J_{\Omega _{P_{S}}} \). Since we have \[ [J_{\Omega _{P_{S}}},T(P_{S},P_{S'})]=0 \] (See (6.2) of \cite{Araki-CCR-2}), the following relation holds : \begin{equation} J_{\Omega _{P_{S}}}\Omega '=\Omega '. \label{eq:T(P,P')x=x} \end{equation} We remark that we have already obtained the following relations : \begin{gather} \label{eq:VQV*=Q'} V^* Q_{P_S}(H)V=Q_{P_{S'}}(H),\\ \label{eq:V*JV=J'} J_{\Omega '} =V J_{\Omega _{P_{S'}}} V^*, \\ \label{eq:JQJ=Q'(Omega)} J_{\Omega _{P_S}} Q_{P_S}(H) J_{\Omega _{P_S}}={\widehat{Q}}_{P_S}(0\oplus H)^{*},\\ \label{eq:JQ(H)J=Q(0+H)(P_S')} J_{\Omega _{P_{S'}}} Q_{P_{S'}}(H) J_{\Omega _{P_{S'}}}=\widehat{Q}_{P_{S'}}(0\oplus H)^* \end{gather} where \( V\) is the unitary operator defined in Lemma \ref{thm:S-S':H.S-->q(S)_q(S'):unitary_eqiv}. We have \begin{equation} J_{\Omega '} Q_{P_S} (H) J_{\Omega '}=\widehat{Q} _{P_S}(0\oplus H)^* \label{eq:JQJ=Q'(Omega')} \end{equation} from \eqref{eq:VQV*=Q'}, \eqref{eq:V*JV=J'} and \eqref{eq:JQ(H)J=Q(0+H)(P_S')}. It follows \begin{gather*} [J_{\Omega '}J_{\Omega _{P_S}}, Q_{P_S}(H)]=0,\\ [J_{\Omega '}J_{\Omega _{P_S}}, \widehat{Q}_{P_S}(0\oplus H)]=0 \end{gather*} for all \( H\in sp(\infty )\) from \eqref{eq:JQJ=Q'(Omega)} and \eqref{eq:JQJ=Q'(Omega')}. Now the center \( C(\mathcal{M}_{P_S} )\) of \( \mathcal{M}_{P_S}\) is trivial : \[ C(\mathcal{M}_{P_S})= \left( \begin{array}{cc} \mathcal{M}^+_{P_S} & 0 \\ 0 & \mathcal{M}^-_{P_S} \end{array} \right) \cap \mathcal{M}'_{P_S}= \left( \begin{array}{cc} C( \mathcal{M}^+_{P_S} ) & 0 \\ 0 & C(\mathcal{M}^-_{P_S} ) \end{array} \right) =\mathbf{C} 1. \] Thus \( J_{\Omega '}J_{\Omega _{P_S}} \in C(\mathcal{M}_{P_S} )=\mathbf{C} 1\), that is, \begin{equation} J_{\Omega '}=\lambda J_{\Omega _{P_S}},\quad \lambda \in S^1:=\{ \lambda \in \mathbf{C} \ |\ |\lambda |=1\}. \label{eq:J=J'} \end{equation} Due to \eqref{eq:T(P,P')x=x} and \eqref{eq:J=J'}, \( \lambda =1\). The second part is verified by \eqref{eq:} and the factoriality of \( \mathcal{M}_{P_{S}}^{+} \) and \[ \left\langle \Omega ',\Omega _{P_{S}}\right\rangle =\left\langle \Omega _{P_{S}},T(P_{S},P_{S'})\Omega _{P_{S}}\right\rangle =\prod _{\lambda \in \sigma (\theta (P_{S},P_{S'}))}\frac{1}{\sqrt{\cosh \lambda }}\geq 0.\] Therefore \( \Omega '\in V_{\Omega _{P_{S}}} \). Now from \eqref{eq:} and Lemma \ref{thm:THEOREM4(8)} and \[\begin{split} P_{S}\cosh \theta (P_{S},P_{S'}) & = P_{S}\sqrt{1-[\sinh \theta (P_{S},P_{S'})]^{2}}\\ & = \sqrt{P_{S}\{1-(P_{S}-P_{S'})^{2}\}}\\ & = \sqrt{P_{S}P_{S'}P_{S}}, \end{split}\] we obtain \eqref{eq:||s(P_S)-s(P_S')||>||w-w'||^2}.\end{proof} \begin{lm}\label{thm:lim||(s-s')|M+||=2} \emph{Assume that \( K \) is separable. Let} \( S,\, S'\in \mathfrak {S} \) \emph{be} \( 0||w-w'||^2}, we can verify this lemma as the proof of Lemma 6.7 of \cite{Araki-CCR-2}. \end{proof} \begin{lm}[Sufficiency of Theorem \ref{thm:Main_Theorem}]\label{thm:Sufficiency_of_Main_Theorem} \emph{Assume that \( K \) is separable}. \( S,\, S'\in \mathfrak {S} \) \textit{and the topologies induced by \( \parallel f\parallel _{S} \) and \( \parallel f\parallel _{S'} \) on \( K \) are equivalent}. \emph{If \( 1-\rho (S)e^{-\chi (S)}e^{\chi (S')}\rho (S') \) is not a Hilbert-Schmidt class operator}, \emph{then \( q_{S}^{\sigma } \) and \( q_{S'}^{\sigma } \) are not quasi-equivalent}.\end{lm} \begin{proof} First suppose that \( 00} d_{P}(A_{\nu },A)\rightarrow 0,\quad d_{P}(B_{\nu },B)\rightarrow 0. \end{equation} \eqref{eq:d(A_n,A)->0} says that for any \( \varepsilon >0 \), if \( \nu \rightarrow \infty \), then \begin{gather} \left\Vert PB_{\nu }(1-P)\right\Vert _{\mathrm{H}.\mathrm{S}.}\leq \left\Vert PB(1-P)\right\Vert _{\mathrm{H}.\mathrm{S}.}+\varepsilon, \label{eq:boundedness_of_net_PB(1-P)} \\ \left\Vert B_{\nu }\right\Vert \leq \left\Vert B\right\Vert +\varepsilon . \label{eq:boundedness_of_net_B} \end{gather} First we claim \begin{equation} \label{eq:convergence_of_PAB(1-P)} \left\Vert P(A_{\nu }-A)B(1-P)\right\Vert _{\mathrm{H}.\mathrm{S}.}\rightarrow 0. \end{equation} \eqref{eq:convergence_of_PAB(1-P)} follows from \eqref{eq:boundedness_of_net_PB(1-P)} and \[\begin{split} \| P(&A_{\nu} -A)B(1-P) \|_{\mathrm{H.S.}}\\ &\le \| P(A_{\nu}-A)(1-P)B(1-P)\| _{\mathrm{H.S.}}+ \| P(A_{\nu} -A)PB(1-P)\| _{\mathrm{H.S.}}\\ &\le \| P(A_{\nu}-A)(1-P)\| _{\mathrm{H.S.}}\|B\| + \|A_{\nu}-A\| \|PB(1-P)\|_{\mathrm{H.S.}} . \end{split}\] Due to \eqref{eq:convergence_of_PAB(1-P)}, we have \( d_{P}(A_{\nu }B_{\nu },AB)\rightarrow 0 \) as \( \nu \rightarrow \infty \). On the other hand, Since \eqref{eq:boundedness_of_net_PB(1-P)}, \eqref{eq:boundedness_of_net_B} and \[\begin{split} \| P(&A_{\nu}^{-1} -A^{-1} (1-P)\| _{\mathrm{H.S.}}\\ &=\| PA_{\nu}^{-1} (A_{\nu}-A)A^{-1}(1-P)\|_{\mathrm{H.S.}}\\ &\le \| PA_{\nu}^{-1} (1-P)(A_{\nu}-A)A^{-1}(1-P)\|_{\mathrm{H.S.}} +\| PA_{\nu}^{-1} P(A_{\nu}-A)A^{-1}(1-P)\|_{\mathrm{H.S.}}\\ &\le \| PA_{\nu}^{-1} (1-P)\|_{\mathrm{H.S.}} \| A_{\nu}-A\| \| A^{-1}\|+ \|A_{\nu}^{-1}\| \|P(A_{\nu}-A)A^{-1}(1-P)\|_{\mathrm{H.S.}}, \end{split}\] we have \( d_{P}(A_{\nu }^{-1},A^{-1})\rightarrow 0 \) as \( \nu \rightarrow \infty \). \end{proof} \begin{lm} \emph{Assume that} \( P_{1} \) \emph{and} \( P_{2} \) \emph{are basis projections and there exist} \( \beta >\alpha >0 \) \emph{such that} \( \alpha \left\Vert f\right\Vert _{P_{1}}\leq \left\Vert f\right\Vert _{P_{2}}\leq \beta \left\Vert f\right\Vert _{P_{1}} \) \emph{for all} \( f\in K \) \emph{and} \( P_{1}-P_{2} \) \emph{is a Hilbert-Schmidt class operator}. \emph{Then} \( Sp(\infty ,P_{1})=Sp(\infty ,P_{2}) \) \emph{as a topological group}. \end{lm} \begin{proof} We have only to show the equivalence of the distance \( d_{P_{1}} \) and \( d_{P_{2}} \). In this proof, we denote the operator norm with respect to \( P \) by \( \left\Vert \cdot \right\Vert _{P} \) and the Hilbert-Schmidt norm with respect to \( P \) by \( \left\Vert \cdot \right\Vert _{\mathrm{H}.\mathrm{S}.,P} \). (In this proof, the Hilbert space norm of \( K \) and the operator norm of bounded operators on \( K \) is same notation, however we probably does not confuse the two meanings.) \[ \left\Vert A\right\Vert _{P_{2}}:=\sup _{x\in K}\frac{\left\Vert Ax\right\Vert _{P_{2}}}{\left\Vert x\right\Vert _{P_{2}}}\leq \sup _{x\in K}\frac{\beta \left\Vert Ax\right\Vert _{P_{1}}}{\alpha \left\Vert x\right\Vert _{P_{1}}}=\frac{\beta }{\alpha }\left\Vert A\right\Vert _{P_{1}}.\] In the same way, we have \( \left\Vert A\right\Vert _{P_{1}}\leq \frac{\beta }{\alpha }\left\Vert A\right\Vert _{P_{2}} \). On the other hand, \[\begin{split} \| P_2 &A(1-P_2)\| _{\mathrm{H.S.},P_2} \\ &\le \beta \| P_2U(P_1/P_2)^{\dagger} U(P_1/P_2)A U(P_1/P_2)^{\dagger} U(P_1/P_2)(1-P_2)\| _{\mathrm{H.S.},P_1} \\ &\le \beta \| U(P_1/P_2)^{\dagger}P_1U(P_1/P_2)A U(P_1/P_2)^{\dagger}(1-P_1)U(P_1/P_2)\| _{\mathrm{H.S.},P_1}\\ &\le \beta \| U(P_1/P_2)^{\dagger}\|_{P_1} \| U(P_1/P_2)\|_{P_1}\\ &\qquad \times \| P_1U(P_1/P_2)A U(P_1/P_2)^{\dagger}(1-P_1)\| _{\mathrm{H.S.},P_1}\end{split}\] and \[\begin{split} \| P_1&U(P_1/P_2)AU(P_1/P_2)^{\dagger}(1-P_1)\| _{\mathrm{H.S.},P_1}\\ &\le \| P_1U(P_1/P_2)\{ P_1+(1-P_1)\} A U(P_1/P_2)^{\dagger}(1-P_1)\| _{\mathrm{H.S.},P_1}\\ &\le \| U(P_1/P_2)\| _{P_1} \| P_1AU(P_1/P_2)^{\dagger}(1-P_1)\| _{\mathrm{H.S.},P_1}\\ &\qquad +\| P_1U(P_1/P_2)(1-P_1)\| _{\mathrm{H.S.},P_1} \| U(P_1/P_2)^{\dagger}\| _{P_1}\| A\|_{P_1}\\ &\le \| U(P_1/P_2)\| _{P_1}\{ \| A\| _{P_1} \| P_1U(P_1/P_2)^{\dagger}(1-P_1)\| _{\mathrm{H.S.},P_1}\\ &\qquad + \| P_1A(1-P_1)\| _{\mathrm{H.S.},P_1} \| U(P_1/P_2)^{\dagger}\| _{P_1} \}\\ &\qquad +\| P_1U(P_1/P_2)(1-P_1)\| _{\mathrm{H.S.},P_1} \| U(P_1/P_2)^{\dagger}\| _{P_1}\| A\|_{P_1}\\ &\le M''(\| A\|_{P_1}+\| P_1A(1-P_1)\| _{\mathrm{H.S.},P_1}) \end{split}\] where \( M'' \) is the maximum value of \begin{gather*} \left\Vert P_{1}U(P_{1}/P_{2})(1-P_{1})\right\Vert _{\mathrm{H}.\mathrm{S}.,P_{1}}\left\Vert U(P_{1}/P_{2})^{\dagger }\right\Vert _{P_{1}}, \\ \left\Vert P_{1}U(P_{1}/P_{2})^{\dagger }(1-P_{1})\right\Vert _{\mathrm{H}.\mathrm{S}.,P_{1}}\left\Vert U(P_{1}/P_{2})\right\Vert _{P_{1}}, \\ \left\Vert U(P_{1}/P_{2})^{\dagger }\right\Vert _{P_{1}}\left\Vert U(P_{1}/P_{2})\right\Vert _{P_{1}}. \end{gather*} Since \( P_{1}U(P_{1}/P_{2})(1-P_{1})=\sinh \theta (P_{1},P_{2})u_{12}(P_{1}/P_{2}) \), \( P_{1}U(P_{1}/P_{2})(1-P_{1}) \) is a Hilbert-Schmidt class operator and \( M'' \) is not infinity. From the above argument, \[ \left\Vert P_{2}A(1-P_{2})\right\Vert _{\mathrm{H}.\mathrm{S}.,P_{2}}\leq M'(\left\Vert A\right\Vert _{P_{1}}+\left\Vert P_{1}A(1-P_{1})\right\Vert _{\mathrm{H}.\mathrm{S}.,P_{1}})\] where \( M':=\beta \left\Vert U(P_{1}/P_{2})^{\dagger }\right\Vert _{P_{1}}\left\Vert U(P_{1}/P_{2})\right\Vert _{P_{1}}M'' \). Let \( M:=M'+\beta \alpha ^{-1} \). Then \( d_{P_{2}}(U_{1},U_{2})\leq Md_{P_{1}}(U_{1},U_{2}) \). In the same way, we can show that there exists a positive number \( m>0 \) such that \( d_{P_{1}}(U_{1},U_{2})\leq \frac{1}{m}d_{P_{2}}(U_{1},U_{2}) \). Therefore \[ md_{P_{1}}(U_{1},U_{2})\leq d_{P_{2}}(U_{1},U_{2})\leq Md_{P_{1}}(U_{1},U_{2}).\] \end{proof} \begin{lm} \emph{Let} \( U\in Sp(\infty ,P) \) \emph{and} \( P':=UPU^{\dagger } \). \begin{enumerate} \item \emph{Let} \( R(U):=U(P/P')U \). \emph{Then} \( R(U) \) \emph{commutes with} \( P \) \emph{and} \( R(U) \) \emph{is an unitary operator on} \( K \). \item \( U(P/P')^{\dagger } \) \emph{is a positive and} 1 + \emph{Hilbert-Schmidt class operator}. \end{enumerate} \end{lm} \begin{proof} (1) Since \( P'=UPU^{\dagger } \) and \( PU(P/P')=U(P/P')P' \), we have \[ PR(U)=U(P/P')P'U=U(P/P')UPU^{\dagger }U=R(U)P.\] Thus \( R(U) \) commutes with \( P \). Due to \( \gamma (R(U)f,R(U)g)=\gamma (f,g) \), we have \[ \gamma _{P}=R(U)^{*}\gamma _{P}R(U)=\gamma _{P}R(U)^{*}R(U).\] ({*} is relative to \( (\cdot ,\cdot )_{P} \).) Since \( P \) is a projection and \( \gamma _{P}^{2}=1 \), we obtain \[ 1=\gamma _{P}^{2}=\gamma _{P}\cdot \gamma _{P}R(U)^{*}R(U)=R(U)^{*}R(U).\] This implies \( R(U)R(U)^{*}=1 \). Thus \( R(U) \) is an unitary operator on \( K \). (2) Due to \eqref{eq:boundedness_HS_norm_of_PU(1-P)}, \( \theta (P,P') \) defined in Lemma \ref{thm:Unitary_Induced_by_Bogoliubov_trans} is a Hilbert-Schmidt class operator. Indeed, we have \begin{equation} \label{eq:H.S.norm_PU(1-P)_equal_tr(PU(1-P)U*P)} \left\Vert PU(1-P)\right\Vert ^{2}_{\mathrm{H}.\mathrm{S}.}=\mathrm{tr}(PU(1-P)U^{\dagger }P) \end{equation} and \begin{gather} PU(1-P)U^{\dagger }P=-[\sinh \theta (P,P')]^{2}P, \label{eq:PU(1-P)U*P} \\ \Gamma \cdot PU(1-P)U^{\dagger }P\cdot \Gamma =-[\sinh \theta (P,P')]^{2}(1-P) \label{eq:PU(1-P)U*P_Gamma} \end{gather} from the direct computation. (\eqref{eq:PU(1-P)U*P_Gamma} follows from \( [\sinh \theta (P,P'),\Gamma ]=0. \)) \eqref{eq:H.S.norm_PU(1-P)_equal_tr(PU(1-P)U*P)}, \eqref{eq:PU(1-P)U*P} and \eqref{eq:PU(1-P)U*P_Gamma} say that \( \sinh \theta (P,P') \) is a Hilbert-Schmidt class operator, i.e. \( \theta (P,P') \) is a Hilbert-Schmidt class operator. We obtain immediately that \( H(P/P') \) is a Hilbert-Schmidt class operator. Since \( iH(P/P') \) is a hermitian operator, the positivity of \( U(P/P') \) is obvious.\end{proof} From the above lemma, \( U\in Sp(\infty ,P) \) is written as \( U=U(P/P')^{\dagger }R(U) \) and this is the polar decomposition of \( U \). We introduce some notations to define the metaplectic representations of \( Sp(\infty ,P) \). Let \( P \) be a basis projection and \( U \) be the element of \( Sp(K,\gamma ,\Gamma ) \) satisfying \( [P,U]=0 \). (Since \( [P,U]=0 \), \( U \) is an unitary operator.) Then the operator \( T_{P}(U) \) on \( \mathcal{H}_{P} \) is defined by \[ T_{P}(U)\pi _{P}(A)\Omega _{P}:=\pi _{P}(\tau (U)A)\Omega _{P},\quad A\in \mathfrak {A}(K,\gamma ,\Gamma ).\] \( T_{P}(U) \) is the second quantization of \( U \). Since \( \tau (U) \) is a {*}-automorphism of CCR algebra \( \mathfrak {A}(K,\gamma ,\Gamma ) \) and \( \varphi _{P} \) is a quasifree state satisfying \[ \varphi _{P}(\tau (U)[B(f)^{*}B(g)])=(Uf,PUg)_{P}=(f,Pg)_{P}=\varphi _{P}(B(f)^{*}B(g)),\] \( T_{P}(U) \) is an unitary operator on \( \mathcal{H}_{P} \). Let \( T_{P}(\Gamma ) \) be an antiunitary operator on \( \mathcal{H}_{P} \) defined by \[ T_{P}(\Gamma )\pi _{P}(A)\Omega _{P}=\pi _{P}(\tau (\Gamma )A)\Omega _{P},\quad A\in \mathfrak {A}(K,\gamma ,\Gamma ).\] \begin{lm} \emph{Let} \( U\in Sp(\infty ,P) \). \emph{Then the unitary operator} \( Q_{P}(U) \) \emph{satisfying} \begin{equation} \label{eq:Bogoliubov_trans_implement_unitary} Q_{P}(U)W_{P}(f)Q_{P}(U)^{*}=W_{P}(Uf) \end{equation} \emph{for all} \( f\in \mathrm{Re}K \) \emph{exists uniquely up to} \( S^{1}:=\{\lambda \in \mathbf{C}\, |\, |\lambda |=1\} \). \end{lm} \begin{proof} Let \[ Q_{P}(U):=T(P,P')T_{P}(R(U)).\] Then \( Q_{P}(U) \) satisfies \eqref{eq:Bogoliubov_trans_implement_unitary}. The uniqueness of \( Q_{P}(U) \) follows from the irreducibility of the von Neumann algebra \( \mathcal{R}_{P}(\mathrm{Re}K) \). In fact, if \( Q'_{P}(U) \) is an another unitary operator satisfying \eqref{eq:Bogoliubov_trans_implement_unitary}, then we have \[ Q'_{P}(U)^{*}Q_{P}(U)W_{P}(f)Q_{P}(U)^{*}Q'_{P}(U)=W_{P}(f)\] for all \( f\in \mathrm{Re}K \) and this shows \[ Q'_{P}(U)^{*}Q_{P}(U)\in \mathcal{R}_{P}(\mathrm{Re}K)'=\mathbf{C}1.\] Therefore \( Q_{P}(U) \) is unique up to the phase factor. \end{proof} \emph{Remark}. Since \( Q_{P}(H) \) defined in the section 4 satisfies \eqref{eq:Q(H)W(f)Q(H)*--projective_case} and \( Q_{P}(U) \) is unique up to the phase factor, we have \( Q_{P}(e^{iH})=\lambda Q_{P}(H) \), \( \lambda \in S^{1} \). Let \[ Q_{P}(\lambda ,U):=\lambda Q_{P}(U)\] for all \( U\in Sp(\infty ,P) \) and \( \lambda \in S^{1} \). \begin{defini} \emph{We denote the group generated by all} \( Q_{P}(\lambda ,U) \) \emph{satisfying} \begin{equation} \label{eq:T(Gamma)_commute_Q_P(lambda,U)} [T_{P}(\Gamma ),Q_{P}(\lambda ,U)]=0 \end{equation} \emph{by} \( Mp(\infty ,P) \). \emph{We call} \( Mp(\infty ,P) \) \emph{the metaplectic group of} \( Sp(\infty ,P) \). \end{defini} The elements \( \lambda \) in \( S^{1} \) satisfying \eqref{eq:T(Gamma)_commute_Q_P(lambda,U)} are \( 1 \) and \( -1 \). In fact, by \( [\Gamma ,U(P/P')]=0 \) and \( [\Gamma ,R(U)]=0 \), we have \( [T_{P}(\Gamma ),T(P,P')]=0 \) and \( [T_{P}(\Gamma ),T_{P}(R(U))]=0 \). This shows \( [T_{P}(\Gamma ),Q_{P}(1,U)]=0 \). Thus \[ T_{P}(\Gamma )Q_{P}(\lambda ,U)=\overline{\lambda }T_{P}(\Gamma )Q_{P}(1,U)=\overline{\lambda }\lambda ^{-1}Q_{P}(\lambda ,U)T_{P}(\Gamma ).\] Due to \eqref{eq:T(Gamma)_commute_Q_P(lambda,U)}, \( \overline{\lambda }\lambda ^{-1}=1 \). Therefore \( \lambda \in S^{1}\cap \mathbf{R}=\{\pm 1\}. \) \begin{prop}\label{thm:property_of_metaplectic_rep} \begin{enumerate} \item \emph{The metaplectic representation \( Mp(\infty ,P) \) is a topological group with respect to the strong operator topology}. \item \emph{The metaplectic representation is continuous projective representation with respect to the topology induced by the distance \( d_{P} \) and the strong operator topology}, \emph{i}.\emph{e}. \emph{if} \( d_{P}(U_{\nu },U)\rightarrow 0 \) \emph{as} \( \nu \rightarrow \infty \), \emph{then} \( Q_{P}(\lambda ,U_{\nu })\rightarrow Q_{P}(\lambda ,U) \) \emph{strongly}. \item \( Mp(\infty ,P) \) \emph{is double covering of} \( Sp(\infty ,P). \) \end{enumerate} \end{prop} \begin{proof} (1) This claim is easily checked. (2) We prove that \( \left\Vert \{Q_{P}(1,U)-1\}\Omega _{P}\right\Vert ^{2}\rightarrow 0 \) if \( d_{P}(U,1)\rightarrow 0 \). Since \[\begin{split} \mathrm{det}_{P}(\cosh \theta (P,P'))^{2} & \leq \exp \left( \left\Vert (\cosh \theta (P,P'))^{2}\right\Vert _{\mathrm{tr}}\right) \\ & = \exp \left( \left\Vert -P(P-P')^{2}P\right\Vert _{\mathrm{tr}}\right) \\ & = \exp \left( \left\Vert PU(1-P)U^{\dagger }P\right\Vert _{\mathrm{tr}}\right) \\ & = \exp \left( \left\Vert PU(1-P)\right\Vert _{\mathrm{H}.\mathrm{S}.}\right) , \end{split}\] we have \[\begin{split} \left\Vert \{Q_{P}(1,U)-1\}\Omega _{P}\right\Vert ^{2} & = 2(1-\mathrm{Re}\left\langle \Omega _{P},T(P,P')\Omega _{P}\right\rangle )\\ & = 2\left\{ 1-\mathrm{det}_{PK}\left( \frac{1}{\sqrt{\cosh \theta (P,P')}}\right) \right\} \\ & = 2\left\{ 1-\frac{1}{^{4}\sqrt{\mathrm{det}_{P}(\cosh \theta (P,P'))^{2}}}\right\} \\ & \leq 2\left\{ 1-\exp \left( -\frac{1}{4}\left\Vert PU(1-P)\right\Vert _{\mathrm{H}.\mathrm{S}.}\right) \right\} . \end{split}\] Thus the first claim has been proved. Moreover, for any \( f\in \mathrm{Re}K \), \[\begin{split} \| \{&Q_P (1,U)-1\} W_P(f) \Omega _P \| \\ &=\| \{ W_P(U^{\dagger} f)Q_P(1,U)-W_P(f)\} \Omega_P\| \\ &\le \| \{ W_P(U^{\dagger} f)Q_P(1,U) -W_P(U^{\dagger}f)\} \Omega _P\| + \| \{W_P(U^{\dagger}f)-W_P(f)\}\Omega_P\|\\ &\le \| \{Q_P(1,U)-1\} \Omega_P \| +\| \{W_P(U^{\dagger}f)-W_P(f)\}\Omega_P\| \end{split}\] From Lemma \ref{thm:Fock-rep-property}(3), if \( \left\Vert U-1\right\Vert \rightarrow 0 \), we have \( \left\Vert \{W_{P}(U^{\dagger }f)-W_{P}(f)\}\Omega _{P}\right\Vert \rightarrow 0 \). We obtain the relation \( \left\Vert \{Q_{P}(1,U)-1\}x\right\Vert \rightarrow 0 \) for all \( x\in \mathcal{H}_{P} \) if \( d_{P}(U,1)\rightarrow 0 \). Thus we have \[ \left\Vert \{Q_{P}(1,U_{\nu })-Q_{P}(1,U)\}x\right\Vert \rightarrow 0\] for all \( x\in \mathcal{H}_{P} \) if \( d_{P}(U_{\nu },U)\rightarrow 0 \). (3) Let \( f_{P}:Mp(\infty ,P)\rightarrow Sp(\infty ,P) \) be a group homomorphism defined by \[ f_{P}(Q_P(\lambda ,U))=U,\quad U\in Sp(\infty ,P),\quad \lambda \in \{\pm 1\}.\] Then \( f_{P} \) is a covering map and \( \ker (f_{P})=\{Q_{P}(-1,1),Q_{P}(1,1)\} \). Thus \[ Mp(\infty ,P)/\ker (f_{P})\simeq Sp(\infty ,P),\] that is, \( Mp(\infty ,P) \) is a double covering of \( Sp(\infty ,P) \). \end{proof} \begin{lm}\label{thm:strong_convergence_of_metaplectic_rep} \emph{Let \( Mp(\infty ,P)_{\mathrm{fin}} \) be the group generated by \( Q_{P}(\lambda ,e^{iH}), H\in sp(\infty )\). Then the closure of \( Mp(\infty ,P)_{\mathrm{fin}} \) with respect to the strong operator topology is \( Mp(\infty ,P) \). That is}, \emph{for any} \( U\in Sp(\infty ,P) \)\emph{,} \emph{there exists a net} \( \{U_{\mu }\} \) \emph{in} \( Sp(\infty ) \) \emph{such that} \[ \mathrm{s}\frac{\, \, }{\, \, }\lim _{\mu \rightarrow \infty }Q_{P}(\lambda ,U_{\mu })=Q_{P}(\lambda ,U).\] \end{lm} \begin{proof} \( T(P,P') \) is written as \[ T(P,P')=\mathrm{s}\frac{\, \, }{\, \, }\lim _{n\rightarrow \infty }Q_{P}(H(P/P')F_{n})=\mathrm{s}\frac{\, \, }{\, \, }\lim _{n\rightarrow \infty }Q_{P}(1,U(P/P')F_{n})\] where \( F_{n} \) is the spectral projection of a positive Hilbert-Schmidt class operator \( \theta (P,P') \) for the open interval \( (\frac{1}{n},\infty ) \). (See the proof of Lemma 5.5 of \cite{Araki-CCR-2}.) On the other hand, since \( R(U) \) is an unitary operator on \( K \), there exists a hermitian operator \( H \) on \( K \) such that \( R(U)=e^{iH} \). Since the set of all finite rank operators on \( K \) is dense set of all bounded operators with respect to the strong {*} operator topology, there exists a net \( \{A'_{\nu }\} \) such that \( A'_{\nu } \) is a finite rank operator and \( A'_{\nu }\rightarrow H \) (strong {*} operator topology) as \( \nu \rightarrow \infty \). Let \( A_{\nu }:=\frac{1}{2}(A'_{\nu }+(A'_{\nu })^{*}) \). Then \( A_{\nu } \) is a finite rank hermitian operator and \( A_{\nu }\rightarrow H \) (strong operator topology) as \( \nu \to \infty \). Let \( H'_{\nu }:=PA_{\nu }P+(1-P)A_{\nu }(1-P) \). Then \( H'_{\nu } \) is a finite rank hermitian operator and commutes with \( P \). Moreover, \( H'_{\nu }\rightarrow H \) (strong operator topology) as \( \nu \to \infty \). In fact, for any \( x\in K \), \[\begin{split} \| (&H'_{\nu}-H)x \| _P\\ &\le \| (PA_{\nu}P-PHP)x\| _P+ \| ((1-P)A_{\nu}(1-P)-(1-P)H(1-P))x\| _P \\ &\le \| (A_{\nu}-H)Px\| _P+ \| (A_{\nu}-H)(1-P)x\| _P\\ &\to 0 (\nu \to \infty ). \end{split}\] Let \( H_{\nu }:=\frac{1}{2}(H'_{\nu }-\Gamma H'_{\nu }\Gamma ) \). Then \( H_{\nu } \) is contained in \( sp(\infty ) \) and satisfies \( [P,H_{\nu }]=0 \) and \( H_{\nu }\rightarrow H \) (strong operator topology) as \( \nu \rightarrow \infty \). This shows \( e^{iH_{\nu }}\in Sp(\infty ) \) and \( \mathrm{s}\frac{\, \, }{\, \, }\lim _{\nu \rightarrow \infty }e^{iH_{\nu }}=R(U) \). Moreover, \[\begin{split} \mathrm{s}\frac{\, \, }{\, \, }\lim _{\nu \rightarrow \infty }T_{P}(e^{iH_{\nu }})W_{P}(f)\Omega _{P} & =\mathrm{s}\frac{\, \, }{\, \, }\lim _{\nu \rightarrow \infty }W_{P}(e^{iH_{\nu }}f)\Omega _{P}\\ & = W_{P}(R(U)f)\Omega _{P}\\ & = T_{P}(R(U))W_{P}(f)\Omega _{P} \end{split}\] for all \( f\in \mathrm{Re}K \). Thus \[ \mathrm{s}\frac{\, \, }{\, \, }\lim _{\nu \rightarrow \infty }\mathrm{Q}_{P}(1,e^{iH_{\nu }})=\mathrm{s}\frac{\, \, }{\, \, }\lim _{\nu \rightarrow \infty }T_{P}(e^{iH_{\nu }})=T_{P}(R(U))=Q_{P}(1,R(U)).\] Now let \( U_{\mu }:=U(P/P')^{\dagger }F_{n}e^{iH_{\nu }} \) where \( \mu =(\nu ,n) \). Then \[ Q_{P}(1,U_{\mu })=Q_{P}(1,U(P/P')^{\dagger }F_{n})Q_{P}(1,e^{iH_{\nu }})\] and \[ \mathrm{s}\frac{\, \, }{\, \, }\lim _{\mu \rightarrow \infty }Q_{P}(1,U_{\mu })=Q_{P}(1,U).\] \end{proof} Let \( Q_{P}^{\sigma }(\lambda ,U) \) be the restriction of \( Q_{P}(\lambda ,U) \) to \( \mathcal{H}_{P}^{\sigma } \) where \( \sigma =+ \) or \( - \). We obtain the following proposition immediately from Lemma \ref{thm:S-S':H.S-->q(S)_q(S'):unitary_eqiv} and Lemma \ref{thm:strong_convergence_of_metaplectic_rep}. \begin{prop} \emph{Suppose that \( K \) is separable. Let} \( P_{1} \) \emph{and} \( P_{2} \) \emph{be basis projections satisfying} \( \alpha \left\Vert f\right\Vert _{P_{1}}\leq \left\Vert f\right\Vert _{P_{2}}\leq \beta \left\Vert f\right\Vert _{P_{1}} \) \emph{for all} \( f\in K \), \( K=K_{P_{1}}=K_{P_{2}} \) \emph{and} \( P_{1}-P_{2} \) \emph{is a Hilbert-Schmidt class operator}. \emph{Then the metaplectic representations} \( Q^{\sigma}_{P_{1}}(\lambda ,*) \) \emph{and} \( Q^{\sigma}_{P_{2}}(\lambda ,*) \) ( \emph{resp . \( Q^{\sigma}_{P_{1}}(\lambda ,*) \) and \( Q^{\sigma '}_{P_{2}}(\lambda ,*) (\sigma \neq \sigma '))\)} \emph{of} \( Sp(\infty ,P_{1})=Sp(\infty ,P_{2}) \) \emph{are unitary equivalent}. (\emph{resp. not unitary equivalent}.) \end{prop} \begin{thebibliography}{99} \bibitem{Araki-CCR-1}Araki, Huzihiro; Shiraishi, Masafumi. {\em On quasifree states of the canonical commutationrelations. I. } Publ. Res. Inst. Math. Sci. 7 (1971/72), 105--120. \bibitem{Araki-CCR-2}Araki, Huzihiro. {\em On quasifree states of the canonical commutation relations.} II. Publ. Res. Inst. Math. Sci. 7 (1971/72), 121--152. \bibitem{Araki-Fock-rep-CAR}Araki, Huzihiro. {\em Bogoliubov automorphisms and Fock representations of canonical anticommutation relations,} in Operator algebras and mathematical physics, 23--141, Contemp. Math., 62, Amer. Math. Soc., Providence, RI, 1987. \bibitem{Radon-Nikodym-theorem}Araki, Huzihiro. {\em Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule.} Pacific J. Math. 50 (1974), 309--354. \bibitem{Borodin-Olshanski}Borodin, Alexei; Olshanski, Grigori. {\em Infinite random matrices and ergodic measures.} Comm. Math. Phys. 223 (2001), no. 1, 87--123. \bibitem{Carey} Carey, A. L.; Ruijsenaars, S. N. M. {\em On fermion gauge groups, current algebras and Kac-Moody algebras.} Acta Appl. Math. 10 (1987), no. 1, 1--86. \bibitem{Lion-Vergne}Lion, G\'{e}rard; Vergne, Mic\'{e}he. {\em The Weil representation, Maslov index and theta series.} Progress in Mathematics, 6. Birkh{\"a}user, 1980. \bibitem{Matsui1} Matsui, Taku. {\em On quasi-equivalence of quasifree states of the gauge invariant CAR algebras.} J.Operator Theory.17,281-290(1987) \bibitem{Matsui}Matsui, Taku. {\em Factoriality and quasi-equivalence of quasifree states for \( Z_{2} \) and \( U(1) \) invariant CAR algebras.} Rev. Roumaine Math. Pures Appl. 32 (1987), no. 8, 693--700. \bibitem{Reed-Simon-2}Michael Reed, Barry Simon. {\em Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. } Academic Press, 1975. \bibitem{Olshanski_Vershik}Olshanski, Grigori; Vershik, Anatoli. {\em Ergodic unitarily invariant measures on the space of infinite Hermitian matrices.} Contemporary mathematical physics, 137--175, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence,RI, 1996. \bibitem{K.R.Parthasarathy}Parthasarathy, K. R. {\em An introduction to quantum stochastic calculus.} Birkh{\"a}user Verlag, 1992. \bibitem{Segal} Pressley, Andrew; Segal, Graeme. {\em Loop groups.} Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. \bibitem{Pickrell}Pickrell, Doug. {\em Separable representations for automorphism groups of infinite symmetric spaces. } J. Funct. Anal. 90 (1990), no. 1, 1--26. \bibitem{Powers}Powers, Robert T. {\em Self-adjoint algebras of unbounded operators.} Comm. Math.Phys. 21 1971 85--124. \bibitem{Stratila-Voiculescu}Str\u{a}til\u{a}, \c{S}erban; Voiculescu,Dan. {\em On a class of KMS states for the unitary group \( \mathrm{U}(\infty ) \).} Math. Ann. 235 (1978), no. 1, 87--110. \end{thebibliography} \end{document} ---------------0204092314929--