% This is a term paper written for Mathematics 277 taught by
% Prof. Alan Weinstein at Berkeley in the Spring of 1997.
%
% The file below is in AMS-LaTeX2e format.
%
%
\documentclass[11pt,draft]{amsart}
%\usepackage{times}
%\usepackage{normsize}
\evensidemargin 0in
\oddsidemargin 0in
\textwidth 6.5truein
\topmargin -0.5truein
\textheight 9truein
\usepackage{tabls}
\newtheorem{theorem}[subsection]{Theorem}
\def\Out{\operatorname{Out}}
\def\Inn{\operatorname{Inn}}
\def\Aut{\operatorname{Aut}}
\def\Tr{\operatorname{Tr}}
\def\End{\operatorname{End}}
\def\id{\operatorname{id}}
\DeclareMathSymbol{\wwwtilde}{\mathalpha}{operators}{"7E}
\author[d. Shlyakhtenko]{Dimitri Shlyakhtenko}
\address{Department of Mathematics, University of California,
Berkeley, CA 94720}
\email{shlyakht@math.berkeley.edu}
\title[$W^*$-algebras and Poisson manifolds]{Von Neumann
Algebras and Poisson Manifolds \\ Term paper for Math 277
}
\date\today
\thanks{The material in this paper is based upon work supported under a
National Science foundation Graduate Research Fellowship. \\ Copyright
\copyright 1997 Dimitri Shlyakhtenko}
\begin{document}
\begin{abstract}
This paper is devoted to the exploration of a number of similarities
between objects related to Poisson manifolds (Poisson bimodules,
symplectic groupoids, etc.) and objects related to von Neumann
algebras (correspondences, standard representations, etc.)
\end{abstract}
\maketitle
\section{Introduction.}
This paper is devoted to the exploration of a remarkable similarity
between Poisson manifolds and von Neumann algebras, a program largely
due to A. Weinstein. As he observed, many objects in von Neumann
algebra theory --- the algebras themselves, bimodules, as well as
operations involving them, such as tensor products --- have analogs in
the world of Poisson manifolds. We attempt to give in this paper a
(certainly far from complete) list of such analogies. A principal
open question that arises from this paper is whether there exists a
functorial correspondence between a certain category of Poisson
manifolds and the category of von Neumann algebras. But even if such
a functorial correspondence were to fail to exist, it is the author's
hope that the use of some ideas from von Neumann algebra theory in
Poisson geometry, and vice-versa, would be beneficial to both
subjects.
One may ask why the category of von Neumann algebras is selected as
the ``target'' of such a functor --- why not $C^*$-algebras, or some
kind of ``smooth'' algebras, etc.? While no fully satisfactory answer
exists, we give two reasons. The first is that Poisson algebras
coming from Poisson manifolds naturally satisfy a bicommutant
condition: if $A\subset B$ are Poisson algebras, let $$A'=\{b\in B:
\{b,A\}=\{0\}\}.$$ Then if $\pi: S\to P$ is a surjective Poisson map
(with connected fibers) from a symplectic manifold to a Poisson
manifold, letting $A=\pi^*C^\infty(P)$, and $B=C^\infty(S)$, one has
$A''=A$. This is analogous to the characterization of von Neumann
algebras as weakly closed subalgebras of bounded operators on Hilbert
spaces; by von Neumann's double-commutant theorem, this topological
condition is equivalent to saying that whenever $M\subset B(H)$ is a
von Neumann algebra, then $M'' = M$.
The second reason comes from the fact that any (topological) manifold
$P$ has a natural measure class on it, which we denote $\mu_P$. This
measure class is defined by the requirement that the push-forwards of
Lebesgue measure on ${\mathbb R}^n$ by all charts are absolutely
continuous with respect to that measure class (i.e., if a set has
measure zero for any (hence every) measure in the measure class, then
it has measure zero with respect to any push-forward of Lebesgue
measure by a chart). Similarly, von Neumann algebras can be thought
of as ``non-commutative Borel spaces with chosen measure classes'';
this is certainly the case for commutative von Neumann algebras
$L^\infty(X,\mu)$ (since every normal state on this algebra
corresponds to a measure absolutely continuous with respect to $\mu$);
for general von Neumann algebras, this has to do with Connes'
Radon-Nikodym type theorem (see below), or, if you like, is just the
existence of the standard form of a von Neumann algebra (also see
below). By contrast, $C^*$-algebras don't have anything like a chosen
measure class; $C(X)$ being the commutative example. (A good
non-commutative example is the CAR algebra, see
\cite{araki,powersstormer,powers:factors,araki-woods,connes}).
For the remainder of the introduction, let us assume that there
actually were a functor $F$ from the category of Poisson manifolds and
certain maps between them to the category of von Neumann algebras,
which satisfied the following properties:
\begin{enumerate}
\item If $S$ is a connected simply connected symplectic manifold, then
$F(S)$ is isomorphic to the algebra $B(H)$ of bounded linear
operators on some Hilbert space $H$. If moreover $L\subset S$ is a
Lagrangian sub-manifold, then there exists a rank-one projection
operator $F(L)\in F(S)$.
\item If $P$ is a Poisson manifold, and its foliation by symplectic
leaves is ergodic (i.e., there are no non-constant functions $f\in
L^\infty(P,\mu)$, with the property that $f$ is invariant under
every Hamiltonian flow), then $F(P)$ is a factor, i.e., the center
$Z(F(P))$ is just multiples of the identity.
\item If $f: S\to P$ is a surjective complete Poisson map from a
connected simply connected symplectic manifold $S$ onto a Poisson
manifold $P$, and $f$ has simply connected fibers, then $F(f):
F(P)\to F(S)$ is a representation.
\item If $f_1 : S\to P_1$ and $f_2: S\to P_2$
are two such maps, and $f_i$ Poisson-commute (i.e., the pull-backs
$f_i^* C^\infty(P_i)$ Poisson commute inside $C^\infty(S)$ endowed
with the canonical Poisson structure), then $F(f_i)(F(P_i))$ commute
in $F(S)\cong B(H)$, and conversely. In particular, if in the above
example, $P_i$ form a dual pair, i.e., $f_i^* C^\infty(P_i) ' =
f_j^* C^\infty(P_j)$, $i\neq j$, $i,j=1,2$, in $C^\infty(S)$, then
$F(f_i)(F(P_i))'=F(f_i)(F(P_j))$.
\item $F$ behaves correctly with respect to the operation of relative
tensor products of pairs (Poisson bimodule, specified Lagrangian
submanifold) and the relative tensor product of the pairs
(correspondence, specified vector in the correspondence).
\end{enumerate}
One candidate for such a functor is the von Neumann algebra associated
canonically to the symplectic groupoid of an integrable Poisson
manifold; another possibility is to consider instead the foliation
algebra of the symplectic foliation of a Poisson manifold. We
consider the extent to which the above conditions are satisfied later
in the paper.
Using $F$ we could make the following ``dictionary'':
\medskip
\begin{tabular}{|l|l|}
\hline
\begin{minipage} {3in} \begin{center}
Poisson geometry \end{center}
\end{minipage} & \begin{minipage}{3in}\begin{center}
von Neumann algebras \end{center} \end{minipage} \\ \hline[1pt]\hline
Poisson manifold $P$
& von Neumann algebra $F(P)$ \\[3pt] \hline
\begin{minipage}{3in}
Complete surjective Poisson map $f:S\to P$ from a connected simply
connected symplectic $S$ onto $P$ \end{minipage} &
\begin{minipage}{3in} Representation $F(f)$ of $F(P)$ on $F(S)$
($\cong B(H)$) \end{minipage} \\ \hline[3pt]
\begin{minipage}{3in} Poisson bimodule $S$, where $S$ is a
connected simply connected symplectic manifold, $f_1$, $f_2$
are complete surjective Poisson maps onto $P_1$, $P_2^o$
(opposite), and $f_i$ Poisson commute \end{minipage} &
\begin{minipage}{3in} $F(S)$ viewed as a left $F(P_1)$ module
and a right $F(P_2)$ module. Since the representations of
$F(P_i)$ commute, this is a correspondence (see e.g.
\cite{connes:correspondences,popa:correspondences}) between
the von Neumann algebras $F(P_1)$ and $F(P_2)$\end{minipage}
\\ \hline[3pt] \begin{minipage}{3in} A Morita equivalence,
or dual pair, between $P_1$ and $P_2$ (see
\cite{ralf:morita}) \end{minipage} & \begin{minipage}{3in}
Index $1$ correspondence (or Morita equivalence) between
$F(P_1)$ and $F(P_2)$ (in this case the images of $F(P_i)$
inside $F(S)$ are commutants of one another).\end{minipage}
\\ \hline[3pt]
\end{tabular}
\medskip
We remark that our requirements on $F$ also imply that
$F(\text{point})={\mathbb C}$. Indeed, since the manifold consisting of
one point is a symplectic connected simply connected manifold, $F$
assigns to it $B(H)$ for some $H$. But the symplectic groupoid of the
one-point manifold is that manifold; moreover, the right and left
projection maps are the same, implying that there exists a (necessarily
faithful) representation of $F(\text{point})$, in which
$F(\text{point})$ is the commutant of itself. But the only $B(H)$ for
which this is possible is $B({\mathbb C})$.
Assume now that $P$ is an integrable Poisson manifold with an
$\alpha$-simply connected symplectic groupoid $S(P)$ (i.e., the fibers
of the source and target maps are simply-connected). Then the
symplectic groupoid can be viewed as a Poisson $(P, P)$ bimodule.
Suppose now that $f: P\to Q$ is a surjective complete Poisson map onto
a Poisson manifold $Q$. Then $S(P)$ is naturally a $(Q, P)$-bimodule,
if we compose the map $\alpha: S(P)\to P$ with the map $f: P\to Q$.
Hence our functor would yield a correspondence $F(S(P))$ between
$S(Q)$ and $S(P)$. Moreover, the identity section of $S(P)$ is a
Lagrangian submanifold, and would hence give a vector $\xi\in
F(S(P))$. Such a correspondence together with $\xi$ can be
interpreted as a completely positive map $F(f): F(Q)\to F(P)$. We
remark that this observation allows one to extend an arbitrary
functor $F$ with the above properties to a functor from the category
of (integrable with $\alpha$-simply connected symplectic groupoid)
Poisson manifolds and surjective complete Poisson maps, to the
category of von Neumann algebras and completely positive maps. It is
not clear if $f\mapsto F(f)$ can be extended to arbitrary Poisson
maps; note in particular that such an extension would automatically
imply the existence of a (rather mysterious!) completely positive map
from $F(P)$ to $F(S)$, where $S$ is a symplectic leaf of $P$.
The rest of the paper is divided as follows. In \S\ref{tomita} we
review the notion of the standard form of a von Neumann algebra,
arising from Tomita-Takesaki theory, and then argue that if a Poisson
manifold $P$ has a symplectic groupoid $S(P)$, then the source and
target maps $\alpha$, $\beta$ from $S(P)$ to $P$, together with the
groupoid structure of $S(P)$ can be viewed as an analog of the
standard form. We then turn in \S\ref{correspondences} to the theory
of correspondences of von Neumann algebras. Next in \S\ref{twoone} we
turn to the representation theory of II$_1$ factors, and the
definition of Jones' index and Jones' basic construction. Finally, in
\S\ref{functor} we discuss the extent to which the ``functor''
assigning to an integrable Poisson manifold the von Neumann algebra of
its symplectic groupoid satisfies the properties demanded of the
correspondence $F$ above.
We would like to mention (but not discuss in the remainder or the
paper) one other instance when von Neumann algebras naturally arise
from geometrical objects, namely, in questions related to studies of
actions of discrete lattice subgroups $\Gamma$ of $PSL(2,{\mathbb R})$ on
the upper half-plane $\mathbb H$. The connection between von Neumann
algebras and such structures was discussed in \cite{gdhj} and in
\cite{dvv:problem}. Later, R{\u a}dulescu considered the von Neumann
algebras arising from a $\Gamma$-equivariant form of Berezin's
quantization of $\mathbb H$ (see
\cite{berezin1,radulescu:gamma,radulescu:hecke}). These papers have
led to some interesting results and conjectures in von Neumann
algebras.
The author would like to acknowledge many inspiring and fruitful
conversations with Professor~Alan~Weinstein, which were a basis for
most of the ideas of this paper. We would also like to thank
D.~Roytenberg for many useful conversations.
\section{The standard form of a von Neumann algebra}
\label{tomita}
Most of the results mentioned in this section can be found in
\cite{stratilazsido} and \cite{stratila:modular}. Let $M$ be a von
Neumann algebra, i.e., a weakly closed $*$-subalgebra of the algebra
of bounded operators on some Hilbert space $H$. On $M$, consider the
ultra-weak topology, given by seminorms
$$\rho_{\xi_1,\xi_2,\dots} (x) = \sum_i \langle x\xi_i, \xi_i\rangle$$
for all sequences $\xi_i\in H$ with $\sum_i \|\xi_i\|^2<\infty$. For
simplicity, we shall assume that $M$ is separable, i.e., can be
faithfully represented on a separable Hilbert space. In this case,
there always exists a faithful normal state $\phi$ on $M$, i.e., a
linear functional, continuous for the ultra-weak topology, which is a
state (i.e., $\phi(x^*x)\geq 0$ for all $x\in M$) and is faithful,
i.e., $\phi(x^*x)=0$ implies $x=0$. Consider the Hilbert space
$K=L^2(M,\phi)$, obtained as the completion of $M$ with respect to the
Hilbert space norm $\|x\|_2=\|x\|_2^\phi = \phi(x^*x)$. Then $M$ acts
on $K$ by right and left multiplication, thus yielding
(automatically normal, i.e., ultra-weakly continuous) faithful
representations $\lambda$ of $M$ and $\rho$ of $M^o$ (opposite) on
$K$; moreover, $\lambda(M)\subset \rho(M^o)'$, $\rho(M)\subset
\lambda(M)'$, where $A'$ denotes the commutant of $A$. Hence $K$ is
an $(M, M)$-bimodule. In fact, $\lambda(M)'=\rho(M^o)$. The vector
$$\xi=1\in K=L^2(M,\phi)= \overline{M}^{_{\|\cdot\|_2^\phi}}$$ is
called the cyclic vector for the representation $\lambda$; indeed,
$M\xi$ is dense in $K$. Note that $\xi$ is also cyclic for the
representation $\rho$ of $M^o$, and vice-versa.
Let $S: K\to K$ be the ${\mathbb C}$-anti-linear operator, densely
defined on elements of $M\subset L^2(M,\phi)=K$ by $Sx=x^*$. Then
$S\xi=\xi$. The basic theorem of Tomita theory is:
\begin{theorem}(Tomita) The operator $S$ is closable.
Let $$S=J\Delta^{1/2}$$ be the polar decomposition of
$S$ with $J$ an anti-linear isometry, and $\Delta=S^*S\geq 0$. Then
$$J \lambda(M) J = \rho(M^o)=\lambda(M)';$$ and for all $t\in
\mathbb R$, and $x\in M$, $$\sigma^\phi_t(x) = \Delta^{it} x
\Delta^{-it}\in M,$$ defining a strongly continuous one-parameter
group of automorphisms $\sigma^\phi_t$ of $M$, called the modular
group of $\phi$.
\end{theorem}
One can check that $\phi$ satisfies the so-called KMS condition,
$$\phi(xy) = \phi(y\sigma_{i}(x)),$$ where $\sigma_i$ denotes the
(densely defined) map $x\mapsto \Delta^{-1} x \Delta$.
Another important object we get from this representation is the cone
$P_+=P_+(M) \subset L^2(M,\phi)$, defined as $$P_+=\overline{\{
\lambda(x) \rho(x^*)\xi: x\in M \}}^{_{H}}.$$ For a cone $P\subset
H$, its dual is given by $$P^o=\{\xi\in H: \langle \xi,\eta\rangle\geq
0, \forall \eta\in P\};$$ $P$ is self-dual if $P=P^o$. The cone $P_+$
is self-dual. Another way to get $P_+$ is to consider the closure of
$\Delta^{1/4} \lambda(M_+)\xi$, where $M_+$ denotes the set of
positive elements (ones having the form $y^*y$ for some $y\in M$) of
$M$.
The $4$-tuple consisting of $M$, $H$, $P_+$ and $J$ constructed above
is called the standard form of $M$. It is a canonical construction,
in the following sense (see \cite{haagerup:standard,araki:modular}):
\begin{theorem} \label{standcanon}
(Araki, Connes and Haagerup) Let $(M,H,P_+,J)$ and
$(\hat{M}, \hat{H},\hat{P}_+, \hat{J})$ be standard forms. If $\Phi:
M\to \hat{M}$ is a $*$-isomorphism, then there exists a unique
unitary $u:H\to \hat{H}$, such that:
\begin{itemize}
\item[(a)] $\Phi(x) = uxu^*,\forall x\in M$;
\item[(b)] $\hat{J} = u J u^*$;
\item[(c)] $\hat{P}_+ = u(P_+)$.
\end{itemize}
\end{theorem}
We consider an example. Let $G$ be a discrete group with counting
measure. Then the group von Neumann algebra $L(G)$ is defined as the
von Neumann algebra generated on $H=\ell^2(G)$ by the left translation
operators $u_g$, $g\in G$. It is easily seen that $$x\mapsto \langle
\delta_e, x\delta_e \rangle = \tau(x)$$ defines a faithful normal
state on $L(G)$ (which is in fact a trace, $\tau(xy)=\tau(yx)$ for all
$x,y\in L(G)$). Then $\ell^2(G)$ is easily seen to be the standard
form representation of $L(G)$; in this case, $\Delta=1$, $J$ is group
inversion, extending anti-linearly the mapping $\delta_g \mapsto
\delta_{g^{-1}}$, and the commutant of $L(G)$ is the von Neumann
algebra generated by right translations.
It turns out that it is occasionally necessary to consider more
general objects on von Neumann algebras than states, called weights.
On a commutative von Neumann algebra $L^\infty(X,\mu)$, a state $\phi$
is the same thing as a probability measure on $X$ (since $\phi(1)=1$).
It is however important to consider other measures on $X$, in
particular measures $\nu$ for which $\nu(X)=+\infty$. The
corresponding ``linear functionals'' can no longer be defined on all
of $L^\infty(X,\mu)$, even if one allows $\pm\infty$ as its values.
Indeed, for the standard Lebesgue measure on $\mathbb R$, it is not
clear what number to assign to the integral $\int_{\mathbb R} x dx$,
since we expect $\int_{\mathbb R} (x+1) dx = \int_{\mathbb R} x dx$ by
invariance of the measure, and on the other hand, linearity would
imply that $\int_{\mathbb R} (x+1) dx = \int_{\mathbb R} x dx +
\infty$, etc. However, if $f\in L^\infty(X,\mu)$ is a positive
function, then $\int_X f d\nu$ is always defined (with possibly an
infinite value). This motivates defining a weight $\phi$ on a von
Neumann algebra $M$ as a linear map from the positive cone
$M_+=\{x^*x: x\in M\}$ to the interval $[0,+\infty]$. A good example
of a weight is the usual trace on $B(H)$ for $H$ infinite-dimensional.
Just as for states, one can define the GNS construction for normal
weights (these are weights that satisfy a certain semicontinuity
condition for the weak topology). The key observation is that any
normal weight can be extended to a linear functional defined a
weakly-dense ideal (trace-class operators in the case of $B(H)$).
Then the representation space becomes (the quotient by vectors of
length zero of) the completion of that ideal with respect to the
Hilbert space norm $\|x\|_2 = \phi(x^*x)$, denoted $L^2(M,\phi)$.
In this case we do not necessarily have an inclusion
$M\subset L^2(M,\phi)$, nor does $L^2(M,\phi)$ necessarily contain the
unit of $M$. However, in the case $\phi$ is faithful (see
\cite{stratilazsido} for the definition), Tomita's conjugation $S$ is
still (densely) defined and furnishes one with $J$ and $\Delta$.
Returning to left regular representations of groups, let $G$ be a
locally compact topological group with Haar measure $\lambda$ and
modular function $\Delta(g) = r_g(\lambda):\lambda$, $r$ denoting the
right translation operator. Then $L(G)$ is the von Neumann algebra
generated on $L^2(G,\lambda)$ by the left translation operators $u_g$,
$g\in G$. Once again this is a standard form representation of
$L(G)$; the commutant is right translation operators $$v_g f (h) =
\Delta^{-1/2}(g) f(hg);$$ the modular operator $\Delta$ is pointwise
multiplication by the modular function, and $$Jf (g)=
\Delta(g)^{{1/2}} \overline{f}(g^{-1})$$ for $f\in L^2(G,\lambda)$ and
$g\in G$. This example is responsible for the word ``modular'' in
modular theory. Since $\delta_{e}$ is in general not in $L^2(G)$, the
above choice of the modular operator does not arise from choosing a
faithful state on $L(G)$. In this case one should instead consider a
faithful weight on $L(G)$, and construct Tomita theory for it.
As another example, let $M=B(H)$ for $H$ finite or infinite
dimensional. Every state on $M$ looks like $\phi(x) = \Tr(D x)$, for
a trace-class positive operator $D$. We assume faithfulness of
$\phi$, which is equivalent to $\ker D = \{0\}$ (equivalently, the
spectral measure of $D$ does not have a point mass at $0$). By an
appropriate choice of basis, we can assume that $D$ is given by a
diagonal matrix, with entries $d_1, d_2,\dots$. Then $L^2(M,\phi)$
can be identified with all matrices $(a_{ij})$ with $\dim H$ entries,
with the inner product $$\langle (a_{ij}), (b_{ij}) \rangle = \sum_{ij}
d_i d_j a_{ij} \overline{b_{ij}}.$$ Note that $$\phi(xy) = \Tr(D xy)
= \Tr(y Dx) = \Tr(D y Dx D^{-1}),$$ hence $x\mapsto D^{it} x D^{-it}$
is the modular group of $\phi$. It is not hard to check that $\Delta$ in
this picture is just left multiplication by $D$ on $L^2(M,\phi)$, $S$
is the operation of taking adjoints, and $J = S \Delta^{-1/2}$.
Even more generally (including all of the examples above!), suppose
that $G$ is a topological groupoid with base $X$ (see \cite{renault}
for definitions and notation; some of the results below are from
\cite{hahn1,hahn2}). Assume that we are given a Haar system $d\lambda_x$,
$x\in X$, i.e., a system of measures $d\lambda_x$, supported on
$\alpha^{-1}(x)$, $x\in X$, for which $$x\mapsto \lambda(\phi)(x) =
\int_{\alpha^{-1}(x)} \phi(g) d\lambda_x(g)$$ is continuous for any $\phi\in
C_c(G)$, and for any $g\in G$, $\phi\in C_c(G)$,
$$ \int_{\alpha^{-1}(\alpha(g))} \phi(gh) d\lambda_{\alpha(g)}(h) =
\int_{\alpha^{-1}(\beta(g))} \phi(h) d\lambda_{\beta(g)} (h).
$$ Assume we are also given a measure class $[\mu]$ on the base $X$,
which is quasi-invariant under the action of $G$; this means that the
measures $\nu$ and $\nu^{-1}$, corresponding to $$C_c(G)\ni \phi\mapsto
\int_X \int_{\alpha^{-1}(x)} \phi(g) d\lambda_x(g)$$ and $$C_c(G)\ni
\phi\mapsto \int_X \int_{\alpha^{-1}(x)} \phi(g^{-1})(x)
d\lambda_x(g),$$ are absolutely continuous with respect to one another.
Then there exists a (locally $\nu$-integrable) positive function
$\Delta$, such that $\nu:\nu^{-1} = \Delta$; it turns out that
$\Delta$ depends only on the class of $[\mu]$, and is a continuous
homomorphism from $G$ to the positive reals.
The compactly-supported continuous functions $A=C_c(G)$ form a
convolution algebra under $$\phi*\psi(g) = \int_{\alpha(h)=\alpha(g)}
\phi(h) \psi(h^{-1} g) d\lambda_{\alpha(g)} (h).$$ One has
a linear map $E: C_c(G) \to C(X)$, given by $$E(\phi)(x) =
\phi(e_x),$$ where $e_x$ is the identity over $x$. Note also that
$C(X)$ acts on $A$ in two ways: for $\gamma \in C(X)$, $\phi\in A$,
the elements
$$(\gamma * \phi)(g) = \gamma(\alpha(g))\phi(g),$$ $$(\phi *\gamma)(g)
= \phi(g) \gamma(\beta(g))$$ lie in $C_c(G)$. (The two actions are
consistent with the idea of thinking of $C(X)$ as ``measures supported
on the identity section of $G$ and then using the groupoid convolution
on measures to define the expressions above). Clearly, the two
actions of $C(X)$ commute, hence $A$ is a $(C(X), C(X))$-bimodule. It
is easily seen that $E$ is a bimodule map, if we think of $C(X)$ as a
bimodule over itself acting by pointwise multiplication. The algebra
$A$ also admits an involution, defined by $$\phi^*(g) = D(g)^{-1}
\overline{\phi}(g^{-1}).$$ Then
$$ E(\phi^* *\phi) (x) = \int_{\alpha^{-1}(x)} \overline{\phi(h)}
\Delta(h) \phi(h) d\lambda_x(h) \geq 0.$$ This property and the fact
that $E$ is bilinear as a map from $A$ (with the
$(C(X),C(X))$-bimodule structure defined above) to $C(X)$ (viewed as a
$(C(X),C(X))$-bimodule in the usual way), means that $E$ is a positive
conditional expectation from $A$ to $C(X)$.
Let now $\mu\in [\mu]$ be a fixed choice of a probability measure on
$X$. Then the following linear functional is well-defined on $A$:
$$\rho(\phi) = \int_X E(\phi)(x) d\mu(x) = \int_X \phi(e_x) d\mu(x).$$
Consider $$\rho(\phi*\psi) = \int_X \int_{\alpha(h)=x} \phi(h^{-1})
\psi(h) d\lambda_x(h) d\mu(x)$$ and $$\rho(\psi * \phi) = \int_X
\int_{\beta(h) = x} \phi(h^{-1}) \psi(h) d\lambda_x(h^{-1}) d\mu(x).$$
Note that $\rho$ is positive, being the composition of a positive
linear map $E$ and a positive functional $C(X)\ni \xi\mapsto \int_X
\xi(x) d\mu(x)$. Then one can easily verify that $\rho$ satisfies the
following KMS condition: $$\rho(\phi * \psi) = \rho (\psi * \Delta
\phi\Delta^{-1}).$$
We can consider the inner product on $A$, given by $$\langle \phi,\psi
\rangle = \rho(\phi^* *\psi) = \int_X \int_{\alpha^{-1}(x)}
\overline{\phi}(h) \Delta(h) \psi(h) d\lambda_x(h),$$ which is equal to
$\langle \phi,\psi\rangle_{L^2(G,\nu^{-1})}.$ Hence the GNS
construction for $\rho$ yields the $*$-representation of $A$ on
$L^2(G,\nu^{-1})$, acting by convolving on the left. It is shown in
\cite{hahn2} and \cite{renault} (see Prop.~1.10 on p.~57) that this is
a standard form representation. Once again, $J$ is given by
$\phi(g)\mapsto \Delta^{1/2} \overline{\phi(g^{-1})}$; the modular
operator is pointwise multiplication by $\Delta$. This representation
is called the regular representation of $G$.
Returning to the general von Neumann algebra theory, we mention a
consequence of the canonicity of the standard form representation:
\begin{theorem} (Araki, Connes and Haagerup) Let $(M,H,P_+,J)$ be a
standard form. Then there is a unique representation $g\mapsto u_g$
of the automorphism group of $M$ on $H$, satisfying $J=u_g J
u_g^{-1}$ and $u_g(P_+) = P_+$.
\end{theorem}
The main ingredient in the proof of Theorem~\ref{standcanon} (of which
the above theorem is an easy consequence) is the observation that the
self-dual positive cone $P_+$ of the standard representation is
homeomorphic to the cone $M_*^+$ in the predual $M_*$ of the $M$ via
the map $P_+\ni\xi\mapsto\omega_{\xi,\xi}$, $\omega_{\xi,\xi}(x) =
\langle \lambda(x)\xi,\xi\rangle$ ($M_*$ is the Banach space of
all ultra-weakly continuous functionals on $M$; $M$ is the
Banach-space dual of $M_*$. $M_*^+$ is the cone consisting of those
linear functionals in $M_*$, which are positive; every element of
$M_*$ is a linear combination of $4$ elements of $M_*^+$.) With this
observation in mind, note that any automorphism of $M$ acts on $M_*$
in a natural way; this action can be used to define a unitary on
$P_+$, which can be extended to all of $K$, and which implements the
automorphism.
We finally mention the Radon-Nikodym type theorem of Connes; recall
that for our purposes a weight is an ``unbounded state'':
\begin{theorem} (Connes) Let $\phi_1$, $\phi_2$ be faithful normal
states on a von Neumann algebra $M$. Then there exists a strongly
continuous one-parameter family of unitaries $u_t\in M$ such that
$$\sigma^{\phi_1}_t (u_t xu_t^*) = \sigma^{\phi_2}_t(x).$$
Conversely, if $u_t\in M$ is any family for which $x\mapsto \alpha_t
(x)=\sigma^{\phi_1}_t (u_t x u_t^*)$ is a one-parameter group
(equivalently, $u_t$ is a cocycle for $\sigma^{\phi_1}_t$), there
exists a normal faithful weight $\phi_2$ such that
$\sigma^{\phi_2}_t = \alpha_t$.
\end{theorem}
One consequence of this theorem is that for any von Neumann algebra
one has a canonical homomorphism $$\delta: {\mathbb R}\to
\Out(M)=\Aut(M)/\Inn(M),$$ given by $t\mapsto \pi\circ \sigma^\phi_t$,
where $\pi$ is the quotient map from $\Aut$ onto $\Out$, and $\phi$ is
any normal faithful state (or, more generally, a normal faithful
semi-finite weight). Connes used this homomorphism to introduce a
classification of factors; the reader is referred to \cite{connes}
(but see also \cite{connes:full}) for details. Let us just record
that $M$ is semifinite (i.e., type I or type II) if and only if
$\delta$ is trivial, and is type III otherwise.
Returning to Poisson manifolds, suppose that $S(P)$ is a symplectic
groupoid with base $P$ (see the lecture notes for this course
\cite{weinstein:poisson:notes}, and also \cite{weinstein:symplectic}).
Then $S(P)$ is called a symplectic groupoid of $P$, and $P$ is called
integrable. The manifold $P$ automatically gets a Poisson structure.
Indeed, the left and right endpoint maps $\alpha$ and $\beta$ provide
two injections $\alpha^*$ and $\beta^*$ of $C^\infty(P)$ into
$C^\infty(S(P))$. Since the fibers of $\alpha$ and $\beta$ are
symplectic orthogonal, $\alpha^* C^\infty(P)$ and $\beta^*
C^\infty(P)$ are Poisson commutants of one another. Thus $$(\alpha^*
C^\infty(P))'' = (\beta^* C^\infty(P))' = \alpha^* C^\infty(P).$$
Suppose now that $f,g\in C^\infty(P)$; we try to define $\{f,g\}\in
C^\infty(P)$. Since $\alpha^*$ is injective, it suffices to show that
$\{\alpha^* f, \alpha^* g\}\in \alpha^* C^\infty(P)$, or,
equivalently, that $\{\alpha^* f,\alpha^* g\}$ commutes with $\beta^*
C^\infty(P)$. Suppose that $h\in \beta^* C^\infty(P)$. Then by the
Jacobi identity, $\{ \{\alpha^* f, \alpha^* g\} , h\}$ is expressed as
a sum of two terms, each involving either $\{\alpha^* f,h\}$, or
$\{\alpha^* g, h\}$, both zero by assumption on $h$. Therefore we
obtain a well-defined operation $\{\cdot,\cdot\}$ on $C^\infty(P)$; it
satisfies all the properties of the Poisson bracket, since the bracket
on $C^\infty(S(P))$ does. Note that $\alpha$ by definition is a
Poisson map. Let $J: C^\infty(S(P))\to C^\infty(S(P))$ be the
anti-Poisson map coming from the groupoid inversion on $S(P)$; then
$J$ intertwines $\alpha$ and $\beta$. Hence we immediately obtain
that $\beta$ is an anti-Poisson map. and
$$J(\alpha^* C^\infty(P)) = \beta^* C^\infty(P) = (\alpha^*
C^\infty(P))'.$$ Thus $S(P)$ is a $(P, P)$-Poisson bimodule.
Recall that according to our ``dictionary'' $F$, the map $\alpha$
(resp. $\beta$) from $S(P)$ onto $P$ should be viewed as a
representation (resp. anti-representation) of $F(P)$ onto a $F(S(P))$.
The fact that images of $\alpha^*$ and $\beta^*$ are Poisson
commutants of one another are thus the analogs of the left and right
actions of a von Neumann algebra $M$ on its standard representation
space $L^2(M,\phi)$; and the properties of the map $J$ we defined on
$C^\infty(S(P))$ above, are analogs of the fact that the map $x\mapsto
JxJ$ is an anti-isomorphism of $M$ onto $M'$ in the standard
representation of $M$.
It is less clear what is the analog in Poisson manifolds of the
positive self-dual cone $P_+$ of Tomita theory, which as we saw above
is a canonical object in von Neumann algebra theory. It seems that
its interpretation should have something to do with
``positive-definite functions'' on the symplectic groupoid $S(P)$.
Indeed, let $G$ be a discrete abelian group. Recall that a function
$f$ on $G$ is positive-definite, if $\sum_1^n f(g_j^{-1}
g_i)\overline{c}_j c_i \geq 0$, for all $g_i\in G$, $c_i\in \mathbb C$
and $n\geq 1$. This is equivalent to requiring that the matrix
$(f(g_j^{-1} g_i))_{ij}$ be positive (as an element of $M_{n\times
n}({\mathbb C})$) for all $g_i\in G$. Then the left regular
representation of the group von Neumann algebra $L(G)$ on $\ell^2(G)$ can
be conjugated by the Fourier transform into the representation of
$L^\infty(\hat{G})$ (the Pontrjagin dual) on $L^2(\hat{G})$. The
positive cone of the representation of $L^\infty(\hat{G})$ consists
precisely of those $f\in L^2(\hat{G})$, which are positive. Hence the
positive cone in $\ell^2(G)$ consists of the Fourier transforms of
positive functions on $\hat{G}$, i.e., by Bochner's theorem, of
positive-definite functions in $\ell^2(G)$.
Nonetheless, one is able to a certain extent recover the statement
about the canonical representation of the group of automorphisms of a
von Neumann algebra on the Hilbert space of its standard
representation. Suppose that $(S(P), \alpha,\beta)$ is a symplectic
groupoid with base $P$, and $\pi: P\to P$ is a Poisson diffeomorphism.
Then (see lecture notes \cite{weinstein:poisson:notes} and that also
\cite{weinstein:symplectic}) the Lie algebroid of $S(P)$ can be
identified with the bundle $T^*P\to P$ in such a way that
$\tilde{\pi}$ is the anchor, and the Lie algebroid bracket is given by
a certain natural expression. This naturality implies that if we
consider $\hat{\pi}: T^*P \to T^*P$, given by $(\omega, p)\mapsto
(\pi_*\omega, \pi(p))$ for $p\in P$, $\omega\in T^*_p P$, then
$\hat{\pi}$ automatically becomes a Lie algebroid map. Depending on
the properties of $S(P)$ (e.g., on whether the fibers of $\alpha$
(equivalently, $\beta$) are simply-connected), it may be possible to
integrate the Lie algebroid map $\hat{\pi}$ to an isomorphism $S(\pi)$
of symplectic groupoid $S(P)$; this extension is quite parallel to the
above canonical implementation of $\Aut(M)$ on $L^2(M,\phi)$. Indeed, note
that $S(\pi)$ would commute with the inversion of $S(P)$ (analog of
commutation with $J$), and would intertwine $\alpha$ and
$\alpha\circ\pi$ (and similarly for $\beta$). This also justifies
denoting by $J$ (rather than by $S$, as the considerations related to
the groupoid algebra would suggest) the Poisson anti-automorphism of
$C^\infty(S(P))$ coming from groupoid inversion on $S(P)$, since $J$
(and not S) is canonical by the considerations above.
We end this section by mentioning the recent work of Weinstein,
interpreting the KMS condition for Poisson manifolds,
\cite{weinstein:KMS}. In particular, Weinstein defined for each
Poisson manifold its modular class, which is a Poisson analog of the
modular homomorphism $\delta: {\mathbb R}\to \Out(M)$. He could then
classify Poisson manifolds into semifinite and type III, by whether
their modular class is trivial or not. For type III Poisson
manifolds, he found an analog of the Connes-Takesaki crossed product
decomposition. We mention also that the KMS condition in Poisson
geometry has been considered in
\cite{lichnerowicz:deform,lichnerowicz:KMS}.
\section{Correspondences and Poisson bimodules}
\label{correspondences}
Let $N$ and $M$ be von Neumann algebras. A correspondence between $N$
and $M$ (or an $(N, M)$-correspondence) is a Hilbert space $H$
admitting a normal unital left representation $\lambda$ of $N$ and a
normal unital right representation $\rho$ of $M$, such that
$\lambda(N)$ and $\rho(M)$ commute in $B(H)$. This definition has
been first considered by Connes in his unpublished notes
\cite{connes:correspondences}; see also Connes' book
\cite{connes:ncgeom}, and the unpublished notes of Popa,
\cite{popa:correspondences}, which we follow in this section.
We say that two correspondences $N,M, H$ and $N,M, H'$ are equivalent,
if there exists a unitary $u:H\to H'$, which is a bimodule map. If
$M$ is a von Neumann algebra, then all of its standard form
representations $L^2(M,\phi)$ for the various choices of $\phi$ are
$(M,M)$-correspondences. The canonicity of the standard form
representation implies that all of these correspondences (regardless
of the choice of $\phi$) are equivalent; we write $L^2(M)$ for the
equivalence class of these correspondences. This is called the
identity correspondence from $M$ to $M$ (see below for an explanation
of the name). If $H$ is an $(N, M)$-correspondence, and
$\hat{M}\subset M$ is a unital subalgebra, then $H$ is naturally an
$(N, \hat{M})$-correspondence, by restricting $\rho$ to $\hat{M}$. In
particular, for any $M$, there are naturally an $(M, {\mathbb
C})$-correspondence, and a $({\mathbb C}, M)$-correspondence, obtained by
restricting the identity correspondence to $\mathbb C$, viewed as a
subalgebra of $M$. As another example, given any $N$, $M$, one can
consider the coarse correspondence $L^2(N)\otimes_{\mathbb C} L^2(M)$,
with the obvious left and right actions. Lastly, suppose $H$ is an
$(N, M)$-correspondence, and $\alpha$ is an automorphism of $N$. Then
one can define the ``twisted'' correspondence $_{\alpha}H$ by
composing the left action $\lambda$ with $\alpha$. In particular, if
$N=M$ and $H=L^2(M)$, then we can naturally associate the
correspondence $H(\alpha) = \; _\alpha L^2(M)$ to $\alpha$. Even more
generally, if $\alpha: K\to N$ is a unital $*$-homomorphism, and $H$
is an $(N, M)$-correspondence, then composing $\lambda$ with $\alpha$
turns $H$ into a $(K, M)$-correspondence.
To explain why correspondences are ``generalized morphisms'', consider
an example where $N=M=L^\infty[0,1]$ with Lebesgue measure. Consider
projections $\pi_1$, $\pi_2$ from $[0,1]\times [0,1]$ onto its first
and second factors. Let $\mu$ be a measure on $[0,1]\times [0,1]$,
such that the push-forwards $(\pi_i)_* \mu$ are absolutely continuous
with respect to Lebesgue measure. Let $H=L^2([0,1]\times [0,1],
\mu)$. Then we have normal representations $\lambda$, $\rho$ of
$L^\infty[0,1]$ given by $$\lambda(f) \xi (x,y) = f(x) \xi(x,y), \quad
\rho(f)\xi(x,y) = f(y) \xi(x,y).$$ It is clear that $H$ is a
correspondence from $M$ to $M$. The case when $\mu$ is the ``delta
measure along the diagonal'' (i.e., integration with respect to $\mu$
corresponds to $f\mapsto \int_0^1 f(x,x) dx$) gives the identity
correspondence; the case when $\mu$ is the product measure gives the
coarse correspondence. If $T:[0,1]\to [0,1]$ is an invertible Borel
transformation, preserving the absolute continuity class of Lebesgue
measure, then one can consider the measure $\mu$ corresponding to
$f\mapsto \int_0^1 f(x,Tx) dx$ (i.e., a ``delta measure along the
graph of $T$''). In this case, we recover the correspondence
$H(\alpha)$, where $\alpha$ is the automorphism of $L^\infty[0,1]$
induced by $T$.
Returning now to the more general situation, let $N$, $M$ be II$_1$
factors, meaning that $N$ and $M$ admit normal faithful
states $\tau_N$ and $\tau_M$ which are traces: $\tau(xy) = \tau(yx)$.
The constructions below can be carried out in more general situations,
but we restrict to the II$_1$ case for the sake of simplicity. We
write $L^2(N)$ for $L^2(N,\tau_N)$, etc.
Suppose that $\eta: N\to M$ is a normal completely positive map, which
means that $\eta$ is ultraweakly-continuous, and $\hat{\eta} :
N\otimes M_{n\times n} \to M\otimes M_{n\times n}$ acting by $\eta$
entry-wise on matrices maps positive elements to positive elements,
for all $n\geq 1$. Completely-positive maps can be also characterized
by saying that these are precisely the maps for which there exists a
Hilbert space $K$ and injections $N\subset B(K)$, $M\subset B(K)$, and
and a bounded linear operator $T:K\to K$, such that $\eta$ is the
restriction of $x\mapsto TxT^*$ to $N\subset B(K)$.
Completely-positive maps naturally arise as ``limits'' of
homomorphisms (a good example is once again the CAR algebra, see
\cite{kadison}, or its free analogs,
\cite{dvv:free,shlyakht:quasifree:big}). This is roughly analogous to
the statement that nonunitary isometries can be obtained as weak
limits of unitaries on a Hilbert space.
Define on the algebraic tensor product $N\otimes M$ the sesquilinear
form $$\langle y_1\otimes y_2, x_1\otimes x_2\rangle = \tau_M
(\eta(y_2^*y_1) x_1 x_2^*).$$ Let $H$ be the Hilbert space completion
of the quotient of $N\otimes M$ by vectors of length zero. Then one
can prove that $H$ is an $(N, M)$-correspondence. When $\eta=\tau_N$,
we get the coarse correspondence; if $\eta$ is the identity map from
$N$ to $M=N$, we get the identity correspondence (explaining its name);
and finally if $\eta$ is an automorphism from $N$ to $M=N$, we get
$H(\alpha)$.
Let $\xi = 1_N \otimes 1_M \in H$, and assume $\eta$ is a
homomorphism. For $x\in N$, we have $$\lambda(x)\xi = \xi \otimes 1\in
H.$$ But \begin{eqnarray*}
\langle x\otimes 1 - 1\otimes \eta(x), x\otimes 1 - 1\otimes
\eta(x)\rangle & = &\tau_M (\eta(x^*x)) + \tau_M(\eta(x) \eta(x^*))\\ & &-
\tau_M(\eta(x) \eta(x^*)) - \tau_M( \eta(x^*) \eta(x)) =
0\end{eqnarray*}
if $\eta$
is a homomorphism. Hence knowing $H$ together with the position of
$\xi\in H$ allows us to recover $\eta$, using the property $$\lambda(x) \xi =
\rho(\eta(x))\xi$$ (provided that knowing $\rho(y)\xi$ completely
determines $y$, which is true, e.g., if $\xi$ is cyclic for $\lambda(N)$).
More generally, suppose $H$ is an $(N, M)$-correspondence. Assume
$\xi\in H$ is of norm $1$ and such that $$\langle \xi\rho(x^*x)
,\xi\rangle \leq c \tau_M(x^*x)$$ for all $x\in M$ (any vector can be
approximated by vectors $\xi$ with such a property). Consider $T:
L^2(M)\to H$ given by $$T(x) = \rho(x)\xi,\quad x\in M;$$ the above
property of $\xi$ assures that $T$ can be extended to a bounded
operator from $L^2(M)\to H$. Then let $$\eta(y) = T^* \lambda(y) T$$
be a map from $N$ to $M$, which is clearly normal and completely
positive. Then the map from the correspondence associated to $\eta$
to the subcorrespondence of $H$
$$\overline{\rho(M)\lambda(N)\xi}^{_H},$$ given by $$x\otimes y\mapsto
x\xi y$$ is an equivalence of correspondences. Although $H$ may fail
to have a vector $\xi$ such that $\rho(M)\lambda(N)\xi$ is dense in
$H$, every correspondence is a direct sum of such ``cyclic''
correspondences.
Let us note that a (cyclic) correspondence between $M$ and $N$ can
be thought of as a specification of an ``absolute continuity class of
maps from $M$ to $N$''. For example, take the case of a
correspondence $H$ between $M$ and $N=\mathbb C$. Then the choice of
a cyclic vector in $H$ corresponds to choosing a state on $M$
(equivalently, a completely positive map $M\to\mathbb C$). When $H$
arises from a state $\phi$ on $M$ via the GNS construction, all the
choices of vectors in $H$ would correspond to choices of states on
$M$, which are ``absolutely continuous'' with respect to $\phi$. For
example, viewing the identity $(M, M)$-correspondence $L^2(M)$ as an
$M$, $\mathbb C$ correspondence, explains to a certain degree why
various vectors in $L^2(M)$ correspond to the various normal
($=$absolutely continuous with respect to a faithful state) states on
$M$ (compare with the identification of $P_+$ with a cone in the
predual $M_*$ of $M$ above).
Similarly, if $\alpha\in \Aut(M)$, then $_\alpha L^2(M)$ depends only
on the class $$[\alpha]\in \Out(M)=\Aut(M)/\Inn(M).$$ Only knowing the
correspondence $_\alpha L^2(M)$ together with a special vector in it
allows one to recover $\alpha$ fully.
We mention (without going into details of the definitions) that there
exists a natural operation of relative tensor products of
correspondences, assigning to an $(N, M)$-correspondence $H$ and an
$(M, K)$-correspondence $\hat{H}$ an $(N, K)$-correspondence
$H\otimes_{M} \hat{H}$.
Lastly, the space of equivalence classes of correspondences between
$N$ and $M$ can be endowed with a natural topology (see
\cite{popa:correspondences} and references therein), which is the
analog of the weak topology on the space of equivalence classes of
representations of a topological groups. This topology can in
particular be used to define property $T$ for von Neumann algebras
(see \cite{connes:propertyT,popa:correspondences}); this property,
just as in the group case, states that the equivalence class of the
identity correspondence (the analog of the trivial representation of
the group) is an isolated point in the space of all equivalence
classes of correspondences.
On the Poisson side of the story, a Poisson $(P, Q)$-bimodule $S$ is a
symplectic manifold $S$ admitting two surjective maps $\alpha: S\to
P$, $\beta: S\to Q$, such that $\alpha$ is a Poisson map and $\beta$
is an anti-Poisson map; further, one assumes that the fibers of
$\alpha$ and $\beta$ are symplectic-orthogonal (which corresponds to
the requirement that $\alpha^*C^\infty(P)$ and $\alpha^* C^\infty(Q)$
commute). We argued above that for an integrable Poisson manifold,
$S(P)$ plays the role of the standard form representation;
equivalently, of the identity correspondence.
If $\pi: P\to P$ is a Poisson diffeomorphism (or more generally, $\pi:
P\to Q$ is a surjective map), one can consider the ``twisted'' Poisson
bimodule $_\pi S(P)$ by composing the map $\alpha$ of the groupoid
with $\pi$ (more generally, one can this way turn any $Q$, $R$ Poisson
bimodule into a $(P, R)$ Poisson bimodule). Just as in the von
Neumann algebra case, one may ask to what extent $_\pi S(P)$
encodes $\pi$.
Clearly, just knowing this bimodule is not enough; for example, let
$\sigma$ be a bi-section of $S(P)$. Then $\sigma$ defines a
diffeomorphism of $P$: for $x\in P$, $\sigma\cdot x$ is
$\alpha(\sigma(x))$. Consider $\pi'(x) = \pi(\sigma\cdot x)$. Then
$_{\pi'} S(P)$ is isomorphic to $S(P)$, simply because $\sigma\cdot
S(P)$ is isomorphic to $S(P)$, where $\sigma\cdot g =
\sigma(\alpha(g)) g$ (groupoid multiplication).
Note that (at least in the neighborhood of the identity
transformation) every Hamiltonian flow ($=$inner automorphism) on $P$
can be realized using the action of a (Lagrangian) bi-section; hence the
above fact that $_\pi S(P)$ is isomorphic to $_{\pi'} S(P)$ can be
viewed as an analog of the von Neumann algebra fact that $_\alpha
L^2(M)$ depends only on the class of $\alpha$ in $\Out(M)$.
Observe that, like the vector $1\in L^2(M)$, $S(P)$ contains a
distinguished Lagrangian submanifold, namely, the identity section
$S(P)^{(0)}$. If we are given $_\pi S(P)$ and the position in it of
the identity section of $S(P)$, we know $\pi$. Indeed, $\pi(x) =
\pi\circ\alpha(g)$, where $g$ is the unique element of $S(P)$,
satisfying $\beta(g)=x$ and $g\in S(P)^{(0)}$; in this way of
recovering $\pi(x)$ we only used the data encoded in $_\pi S(P)$ and
the position of $S(P)^{(0)}$. This is just like saying that knowing
$_\alpha L^2(M)$ and the vector $1$ in this correspondence allows us
to recover $\alpha$.
We remark that as pointed out in \cite{weinstein:symplectic},
Lagrangian submanifolds can be thought of as giving vectors in Hilbert
spaces when quantizing symplectic manifolds. For example, if $T^*M$
is a symplectic manifold and $f\in T^*(M)$, then the graph of $df$ in
$T^*(M)$ is a Lagrangian submanifold, thought to encode $\exp(if)\in
L^2(M)$, which should be the quantization of $T^*M$. However, this
correspondence is far from straightforward (see
\cite{weinstein:symplectic}).
To conclude this section, we pose several questions:
\begin{enumerate}
\item Is there a Poisson analog of the topology on the set of all $(N,
M)$-correspondences? If such an analog is found, it would be
interesting to investigate manifolds with property $T$ (defined in
terms of this topology as in \cite{popa:correspondences}, also in
\cite{connes:correspondences,connes:propertyT}, which mirrors the way
property $T$ is defined in terms of representation theory of
groups), and see if there are any rigidity results for them.
\item Can one understand the analog of $P_+$ for Poisson manifolds
by using the fact that for von Neumann algebras, all normal states
$M_*^+$ correspond to various choices of vectors in $L^2(M)$? It
would appear that the related question for Poisson manifolds would
involve studying all Lagrangian submanifolds of $S(P)$. Is there an
analog of the GNS construction --- does every suitable Lagrangian
submanifold produce a $P$, $\text{pt}$ Poisson bimodule?
\item If $\pi$, $\pi'$ are two surjective Poisson diffeomorphisms of
$P$, and $_\pi S(P)$ and $_{\pi'}S(P)$ are isomorphic as Poisson
bimodules, what can be said of $\pi$ and $\pi'$?
\end{enumerate}
\section{Representation theory of II$_1$ factors}
\label{twoone}
Recall that a von Neumann algebra is a factor if its center is
trivial. Factors are classified into three groups: type I and type
II, possessing faithful traces (i.e., weights (=``unbounded states'')
invariant under all inner automorphisms, and for which the GNS
representation is faithful), and type III, not possessing any traces.
Factors of type I are of the form $B(H)$ for some (finite or infinite
dimensional) $H$; the range of the trace on the set of projections
(i.e., elements $p$ with $p=p^*=p^2$) of a type I factor is discrete.
Type two factors are divided into two subsets, type II$_1$ and
II$_\infty$, according to whether the trace is finite or infinite on
the identity of the algebra.
Thus a II$_1$ factor $M$ is a factor possessing a faithful normal
trace, i.e., a ultra-weakly continuous linear functional $\tau: M\to
\mathbb C$, satisfying, for all $x,y\in M$, $\tau(x^*x)\geq 0$,
$\tau(x^*x)=0$ iff $x=0$, and $\tau(xy)=\tau(yx)$. A II$_\infty$
factor is a von Neumann algebra of the form $N=M\otimes B(H)$, where
$M$ is a II$_1$ factor and $H$ is infinite-dimensional. $N$ has a
semi-finite trace, given by the tensor product $\tau_M\otimes \Tr$;
this trace is not defined on all elements of $N$. It is however
defined on all positive elements (the value $+\infty$ is allowed), in
particular on all projections, i.e., elements $p$ with $p^2=p^*=p$. The
trace $\tau_M$ on a II$_1$ factor is unique, given the normalization
$\tau_M(1_M)=1$; the trace $\tau_N$ on a II$_\infty$ factor is unique
up to a positive constant multiple. The range of the trace restricted
to projections on a II$_1$ factor is the entire interval $[0,1]$; on a
II$_\infty$ factor, it is all of $[0,+\infty]$.
Examples: II$_1$ factors can be produced from discrete groups. If $G$
is a so-called ICC group, i.e., a discrete group such that for any
$e\neq g\in G$, the conjugacy class $\{hgh^{-1} : h\in G\}$ is
infinite, then the group von Neumann algebra $L(G)$ is a II$_1$ factor
(see above for the definition and discussion of $L(G)$). $L(G)\otimes
B(H)$ for infinite-dimensional $H$ is then a II$_\infty$ factor.
Other examples arise from ergodic transformations (see
\cite{krieger}), ergodic equivalence relations (see
\cite{feldman-moore}) and more generally, as von Neumann algebras
generated in the regular representation by a groupoid algebra (see
\cite{hahn2}); these are all generalizations of the original
``group-measure'' construction of Murray and von Neumann. More
examples can be built from these by considering tensor products of
II$_1$ factors. Lastly, examples arise from free products and free
probability
(\cite{dvv:book,DVV:circular,dykema:interpolated,dykema:fdim,%
radulescu:subfact}) and by iterating free products and tensor
products (\cite{ge:entropy2}).
Suppose that $M$ is a von Neumann algebra acting normally on a Hilbert
space $H$. Then Murray and von Neumann proved that if $M$ is type I
(resp. type II, type III), then the commutant $M'$ of $M$ in $B(H)$ is
type I (resp. type II, type III). The commutant of a factor $M$ is
always anti-isomorphic to an amplification of $M$, i.e., the algebra
$p (M\otimes B(H)) p$, where $p$ is some nonzero projection $p\in
(M\otimes B(H))$. In the case $M$ is type III, this algebra does not
depend on $p$, up to isomorphism. Since in a type I or II factor, two
projections are unitarily equivalent if and only if they have the same
(finite or infinite) trace, $p(M\otimes B(H))p$ only depends on the
trace of $p$; it is customary to write $M_\alpha$ for $p(M\otimes
B(H))p$ if $\Tr(p)=\alpha$. We'll write $M_\alpha$ in the type III
case, although, as pointed out above, this doesn't really depend on
$\alpha$ (as long as we stay in the separable situation).
As an application, this gives yet another picture for a correspondence
$H$ between $N$ and $M$ in the case $N$ is a factor: by definition, we
have the right action $\rho: M\to B(H)$ whose range is contained in
the commutant of $N$, which is anti-isomorphic to $N_\alpha$ for some
$\alpha$. Hence we have a unital anti-homomorphism $\rho: M\to
N^o_\alpha$, or a unital homomorphism $\gamma:M\to N_\alpha$.
Conversely, given such a unital homomorphism, $\gamma: M\to N_\alpha$,
construct $H$ as follows. Start with the $( N, N\otimes
B(K))$-correspondence $$L^2(N)\otimes_{\mathbb C} K$$ Then $N_\alpha$
is isomorphic to $$p(N\otimes B(K))p, \quad\text{for some $p\in
(N\otimes B(K))$}.$$ Let $$H = \rho(p) (L^2(N)\otimes_{\mathbb C}
K);$$ then $H$ is an $(N, N_\alpha)$-correspondence. Now using
$\gamma$ we can make this into a $(N, M)$-correspondence, denoted by
$L(\gamma)$, as we discussed before. The equivalence of such
correspondences is just the notion of equivalence on maps $\gamma:
M\to N_\alpha$, where $\gamma: M\to N_\alpha$ is equivalent to
$\gamma': M\to N_{\alpha'}$ if $\alpha=\alpha'$ and for some unitary
$u\in N_\alpha$, $\gamma(x) = u\gamma'(x) u^*$.
These considerations allow one to describe, in a different way, the
operation of tensor product of correspondences. Indeed, suppose
$\gamma: N\to M_\alpha$ and $\gamma': M\to P_\beta$ are unital maps.
Then form $\gamma'_\alpha: M_\alpha\to (P_\beta)_\alpha \cong
P_{\alpha\beta}$ in the obvious way. Then $$L(\gamma)\otimes_M
L(\gamma') = L(\gamma'_\alpha \circ \gamma).$$ This gives
yet another explanation for the term identity correspondence: since
the identity correspondence is just $L(\id)$, we have $L^2(M)
\otimes_M H = H $ for any $(M,N)$-correspondence $H$.
Another application is to representation theory of a II$_1$ factor
$M$. Indeed, classifying representations of $M$ is the same as
classifying $(M, {\mathbb C})$-correspondences up to equivalence; i.e.,
classifying unital maps ${\mathbb C}\to M_\alpha$. These are clearly
classified by $\alpha$. Given a representation of $M$ on $H$,
$\alpha$ (called the coupling constant of $M$ and $M'$, or the
dimension $\dim_M (H)$) can be computed in the following way: let
$\xi\in H$ be a non-zero vector. If $M'$ is type II$_\infty$,
$\alpha=\infty$. Let $p_\xi$ be the orthogonal projection onto the
closure of $M\xi$; let $p_\xi'$ be the orthogonal projection onto the
closure of $M'\xi$. The projection onto an subspace invariant for $A$
is in $A'$. Hence $p_{\xi}\in M'$, $p_{\xi'}\in M$. If $M'$ is type
II$_1$, it has a normalized trace $\tau_{M'}$. Thus the number $$c =
{\frac{\tau_{M'} ( p_\xi )} {\tau_{M} (p_{\xi'})}}$$ is well-defined,
and is actually equal to $\alpha$ (hence independent of the choice of
$\xi$!)
In particular, suppose $N\subset M$ is a subfactor of $M$ (i.e., $N$
and $M$ are factors, and the inclusion is unital). Then $N$ acts on
$L^2(M)$, hence the number $\dim_N(L^2(M))$ is well-defined. This
number was considered by Jones in \cite{jones:index}, and called the
index of $N$ in $M$, denoted $[M:N]$. The motivation for the name is
that if $H\subset G$ is a subgroup, then $[L(G):L(H)] = [G:H]$. This
definition led to a vast body of literature; see e.g. \cite{gdhj}.
One of the main tools is the so-called basic construction. Let
$N\subset M$ be a subfactor. Then define $M_1$ as the von Neumann algebra
generated on $L^2(M)$ by $M$ and the projection $p_N$ onto $L^2(N)$;
$p_N$ is called the Jones projection. Then $M\subset M_1$ and
moreover $[M_1:M]=[M:N] = \tau_{M_1}(p_N)^{-1}$. Hence to an
inclusion $N\subset M$ one can associate, by iterating the basic
construction, the tower of algebras $$N\subset M\subset M_1\subset M_2
\subset\dots,$$ together with distinguished projections $p_i$ (which
are Jones projections for $M_{i-1}\subset M_{i}$). This tower of
algebras has an invariant, namely the (finite dimensional when the index
is finite) algebras $M_i'\cap M_j$, $i\leq j$. Using the relations
between Jones projections, Jones in \cite{jones:index} showed that the
possible values of the index of $N\subset M$ lie in the set
$$\{4\cos^2(\pi/q): q\in {\mathbb N}, q\geq 3\}\cup [4,+\infty].$$ We
refer to \cite{jones:index,gdhj} for an exposition. Finally, let us
mention that the inclusion $N\subset M$ gives a natural bimodule
category (see \cite{bisch:bimodules} for an exposition), by
interpreting $M_i$ as $M_j$ bimodules for various $i$ and $j$. Such
categories lead to topological invariants, such as the Jones
polynomial (see \cite{jones:knots}). Notice that since we needed a
natural way of including $L^2(N)$ into $L^2(M)$, the condition that
$M$ (and hence $N$) are type II$_1$ is essential. For more general
factors, one can define the index of the triple $N\subset M, E:M\to
N$, where $E$ is a conditional expectation (i.e., normal positive
$N$-bilinear map); the index would depend on the choice of $E$ (and in
the II$_1$ case, there is a canonical choice, namely the $E$ that
preserves the trace on $M$). It turns out that in the type II$_1$
case, $L^2(M_1)$ (as an $(M, N)$-correspondence) is the correspondence
associated to the completely positive map $E:M\to N$.
We give a partial list of questions regarding Poisson manifolds, that
naturally arise from the above considerations:
\begin{enumerate}
\item Given a Poisson manifold of type II$_1$ (i.e., possessing a
unique probability measure invariant under all Hamiltonian flows),
is its commutant in every representation type II$_1$ or II$_\infty$?
Is there an analog of the coupling constant?
\item Are the representations (up to a suitable notion of equivalence)
classified by the coupling constant?
\item What is the analog of the amplification $P_\alpha$ of $P$?
For a von Neumann algebra, the statements that $N$ is an
amplification of $M$ and that $N$ is Morita-equivalent to $M$ are
the same, hence one could suggest that $P_\alpha$ should be
Morita-equivalent to $P$.
\item What is the analog of the fundamental group of a von Neumann
algebra $F(M)= \{\alpha\in {\mathbb R} : M\cong M_\alpha\}$?
\item Is there a basic construction for Poisson manifolds? Is there a
notion of Jones projection?
\item Is there a notion of the index? Are there restrictions on the
possible values of the index?
\item More generally, are $(P, Q)$-bimodules, up to a suitable notion
of equivalence, the same as a surjective maps from an
amplification of $Q$ to $P$?
\end{enumerate}
For defining the basic construction, the following remark from
\cite{gdhj} is helpful. Given $N\subset M$, one can define $M_1$ (the
basic construction) as $\End^r_{N}(M)$, i.e., the algebra of
endomorphisms of $L^2(M)$ viewed as a right $N$-module. This allows
one to define, if $P$ is the base of a symplectic groupoid $S(P)$, the
basic construction for Poisson algebras: given a surjective map of
$\pi:P \to Q$, we define the Poisson algebra $P_1$ as the Poisson
commutant in $C^\infty(S(P))$ of $\beta^*\pi^* C^\infty(Q)$. It is
not clear when this corresponds to a manifold, nor whether there is
some analog of the statement that it is generated by $\alpha^*
C^\infty(P)$ and a ``Jones projection''. If we start with the
inclusion ${\mathbb C}\subset M$, then the basic construction gives
$M_1= B(L^2(M))$. Similarly, in the Poisson case, we get for the map
$P\to \text{pt}$ the basic construction $S(P)\to P\to \text{pt}$.
Hence to some extent the construction of $P_1$ (or perhaps more
precisely, $S(P_1)$) from $P\to Q$ should be a ``$Q$-valued analog''
of the construction of $S(P)$ from $P$ (note that the latter
construction is already highly non-trivial, since $S(P)$ is not really
canonically associated to $P$; e.g., the fibers of the different
choices of the groupoid may have different fundamental groups.)
\section{The groupoid von Neumann algebra of a Poisson manifold}
\label{functor}
Suppose that $P$ is a Poisson manifold which possesses a symplectic
groupoid $S(P)$ so that the fibers of $\alpha$ are simply-connected.
Then we can consider the von Neumann algebra $F(P)$ generated in the
left regular representation of the groupoid algebra of $S(P)$ on
$L^2(S(P))$. We mention that Weinstein has in
\cite{weinstein:ncgeom,weinstein:xu} considered a ``twisted'' version of this
algebra, corresponding to the choice of a ``prequantization'' of
$S(P)$.
The correspondence between a groupoid with a chosen measure class on
the base and the von Neumann algebra generated in the left regular
representation by its groupoid algebra is functorial; hence whenever
one exhibits a functorial property of $P\mapsto S(P)$, it will lead to
an associated functorial property of $P\mapsto F(P)$.
We discuss to what extent the properties desired of $F$, listed at the
beginning of the paper, are satisfied.
\begin{enumerate}
\item If $S$ is a simply connected symplectic manifold, we can take
$S\times \overline{S}$ as $S(S)$. The groupoid structure on $S(S)$
is that of a pair groupoid: the identity section is the diagonal,
and the maps $\alpha$, $\beta$ are the two projections onto $S$ and
$\overline{S}$ (isomorphic to $S$ as a manifold, being $S$ with the
negative of the symplectic structure on $S$). As a measure space,
$S$ with the manifold measure is equivalent to the reals $\mathbb R$
with Lebesgue measure (see e.g. \cite{petersen:ergodic}). Hence
the pair groupoid $S(S)$ is isomorphic as a measure groupoid to the
pair groupoid of $\mathbb R$. The groupoid algebra of that can be
identified with the algebra generated by integral kernel operators
$f(x,y)$, $x,y\in \mathbb R$. It is a classical fact (cf.
\cite{edwards:funcanal}) that the $C^*$-algebra these generate is
the compact operators; the only von Neumann algebra that compact
operators can generate in a Hilbert space representation (being
simple) is $B(H)$ for some $H$. The question of how to
associate to a Lagrangian subspace $L\subset S$ a rank-one
projection in $F(S)$ (equivalently, a vector in $H$) has been
discussed before, and the answer is not clear.
\item If the groupoid $S(P)$ happens to be principal (which is
equivalent to the requirement that $P$ be symplectic), then
this is settled by a result of Hahn (see Theorem~5.1 of
\cite{hahn2}). In that case the image of $L^\infty(P)$ inside the
von Neumann algebra $F(P)$ is maximal abelian ($L^\infty(P)'\cap
F(P) = L^\infty(P)$). When $S(P)$ is not principal, the answer is
less clear; in that case $F(P)$ may fail to be a factor even if the
symplectic foliation of $P$ is ergodic.
\end{enumerate}
Whether properties 3, 4, and 5 hold is not clear. Property 3
certainly holds for $P$ a symplectic manifold. All the properties in
4 are satisfied in the case that $S=S(P)$. Because of the
infinitesimal picture for $S(P)$ provided by its Lie algebroid, and
the identification of that Lie algebroid with $T^*(P)$, we do have
that $F(P)$ is a functor from integrable Poisson manifolds and local
diffeomorphisms to von Neumann algebras. However, if $S$ is
symplectic and $S\to P$ is a surjective Poisson map, it is not clear
how to define a map from $T^*P$ to $T^*S$.
The advantage of using the von Neumann algebra of the foliation of a
Poisson manifold by symplectic leafs is that the resulting algebra is
always a factor, if the foliation is ergodic. However, it is not
clear how, given a Poisson map between two Poisson manifolds, to
construct from this map a homomorphism of the associated von Neumann
algebras.
Note also that the symplectic groupoid von Neumann algebra of a
manifold with trivial Poisson structure seems too large (cf.
\cite{weinstein:ncgeom}). We would expect to associate to such a
manifold $P$ the commutative von Neumann algebra $L^\infty(P)$ (for
the manifold measure). However, a choice of the symplectic groupoid
is $T^*P$, with $\alpha$ and $\beta$ being the bundle projection $T^*P
\to P$, and groupoid operations coming from addition and negation on
the fibers of $T^*P$. As a measure groupoid, this groupoid is
isomorphic to the bundle of groups ${\mathbb R}^n$ over $P$, hence,
using the isomorphism $L({\mathbb R}^n)\cong L({\mathbb R})$, its
associated von Neumann algebra is isomorphic to $L({\mathbb R})\otimes
L^\infty(P)$, and not in a natural way to $L^\infty(P)$. One
possibility is the choice of a ``polarization'' of the groupoid
$S(P)$, for the discussion of which we refer the reader to
\cite{weinstein:ncgeom,weinstein:irrat,weinstein:xu}.
On the other hand, as pointed out in \cite{weinstein:KMS}, the modular
theory of Poisson manifolds developed by Weinstein connects nicely
with the modular theory of the von Neumann algebra of a symplectic
groupoid associated to a Poisson manifold.
Finally, we mention that the functor $F$ is cannot be onto. Indeed,
the von Neumann algebra $B$ generated in the left regular
representation of a groupoid contains the algebra $A$ of functions on
the base of the groupoid as a regular subalgebra: the entire von
Neumann algebra is generated by $A$ and the normalizer of $A = \{u\in
B \text{ unitary} : uAu^*\subset A\}$ (which in turn contains all
operators arising from bisections); moreover, in our case the base is
a standard Lebesgue space, so $A\cong L^\infty[0,1]$. But by a recent
result of Voiculescu \cite{dvv:entropy3}, the von Neumann algebra
$L({\mathbb F}_2)$ fails to have diffuse regular hyperfinite
subalgebras, so it cannot arise from such a groupoid algebra.
\bibliographystyle{amsplain}
%\bibliography{../../bib/quasifree,paper}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}
\bibitem{araki}
H.~Araki, \emph{On quasifree states of {CAR} and {Bogoliubov} automorphisms.},
Publ. RIMS Kyoto Univ. \textbf{6} (1970), 385--442.
\bibitem{araki:modular}
\bysame, \emph{Some properties of modular conjugation operator of {von Neumann}
algebras and a non-commutative {Radon-Nikodym} theorem with a chain rule},
Pacific J. Math. \textbf{50} (1974), 309--354.
\bibitem{araki-woods}
H.~Araki and E.~J. Woods, \emph{A classification of factors}, Publ. RIMS Kyoto
Univ. series {A} \textbf{4} (1968), 51--130.
\bibitem{lichnerowicz:deform}
H.~Basart, M.~Flato, A.~Lichnerowicz, and D.~Sternheimer, \emph{Deformation
theory applied to quantization and statistical mechanics}, Letters in Math.
Physics \textbf{8} (1984), 483--494.
\bibitem{lichnerowicz:KMS}
H.~Basart and A.~Lichnerowicz, \emph{Conformal symplectic geometry,
deformations, rigidity and geometrical {(KMS)} considtion}, Letters in Math.
Physics \textbf{10} (1985), 167--177.
\bibitem{weinstein:symplectic}
S.~Bates and A.~Weinstein, \emph{Lectures on the geometry of qunatization},
vol.~8, Berkeley Mathematics Lecture Notes, 1995.
\bibitem{berezin1}
F.~Berezin, \emph{Quantization in complex symmetric spaces}, Math USSR Izvestia
\textbf{9} (1975), 341--379.
\bibitem{bisch:bimodules}
D.~Bisch, \emph{Bimodules, higher relative commutatnts, and the fusion algebra
associated toa subfactor}, Preprint, Berkeley, 1995.
\bibitem{connes:correspondences}
A.~Connes, \emph{Correspondences}, unpublished notes.
\bibitem{connes}
\bysame, \emph{Une classification des facteurs de type {III}}, Ann. scient.
{\'Ec.} Norm. Sup. \textbf{6} (1973), 133--252.
\bibitem{connes:full}
\bysame, \emph{Almost periodic states and factors of type {III$_1$}}, J. Funct.
Anal. \textbf{16} (1974), 415--455.
\bibitem{connes:ncgeom}
\bysame, \emph{Noncommutative geometry}, Academic Press, 1994.
\bibitem{connes:propertyT}
A.~Connes and V.F.R. Jones, \emph{Property {$T$} for von {Neumann} algebras},
Bull. London Math. Soc. \textbf{17} (1985), 57--62.
\bibitem{dykema:fdim}
K.~Dykema, \emph{Free products of hyperfinite von {Neumann} algebras and free
dimension}, Duke Math J. \textbf{69} (1993), 97--119.
\bibitem{dykema:interpolated}
\bysame, \emph{Interpolated free group factors}, Pacific J. Math. \textbf{163}
(1994), 123--135.
\bibitem{edwards:funcanal}
R.~Edwards, \emph{Functional analysis}, Mir, 1969, (in Russian).
\bibitem{feldman-moore}
J.~Feldman and C.~C. Moore, \emph{Ergodic equivalence relations, cohomology,
and von {Neumann} algebras {I}, {II}}, Trans. AMS \textbf{234} (1977),
289--359.
\bibitem{ge:entropy2}
L.~Ge, \emph{Applications of free entropy to finite von {Neumann} algebras,
{II}}, Preprint, 1996.
\bibitem{gdhj}
F.M. Goodman, R.~{de la} Harpe, and V.F.R. Jones, \emph{Coxeter graphs and
towers of algebras}, Springer-Verlag, 1989.
\bibitem{haagerup:standard}
U.~Haagerup, \emph{The standard form of {von Neumann} algebras}, Math. Scand.
\textbf{37} (1975), 271--283.
\bibitem{hahn1}
P.~Hahn, \emph{Haar measure for measure groupoids}, Trans. AMS \textbf{242}
(1978), 1--33.
\bibitem{hahn2}
\bysame, \emph{The regular representations of measure groupoids}, Trans. AMS
\textbf{242} (1978), 35--72.
\bibitem{dvv:problem}
{P. de la} Harpe and D.-V. Voiculescu, \emph{A problem on the {II$_1$}-factor
of {Fuchsian} groups}, Recend advances in Operator algebras (Orl\'eans,
1992), no. 232, Ast\`erisque, 1995, pp.~155--158.
\bibitem{jones:index}
V.F.R. Jones, \emph{Index for subfactors}, Invent. Math \textbf{72} (1983),
1--25.
\bibitem{jones:knots}
\bysame, \emph{A polynomial invariant for knots via von {Neumann} algebras},
Bull. AMS \textbf{12} (1985), 103--112.
\bibitem{kadison}
R.~Kadison, \emph{Notes on {Fermi} gas}, Symposia math \textbf{{XX}} (1976),
425--431.
\bibitem{krieger}
W.~Krieger, \emph{On constructing non-$*$-isomorphic hyperfinite factors of
type {III}}, J. Funct. Anal \textbf{6} (1970), 97--109.
\bibitem{ralf:morita}
R.~Meyer, \emph{Morita equivalence in algebra and geometry}, Term paper for
{Mathematics} 277, UC Berkeley, see {{\tt
/$\wwwtilde$alanw/}}, 1997.
\bibitem{petersen:ergodic}
K.~Petersen, \emph{Ergodic theory}, Cambridge university press, 1983.
\bibitem{popa:correspondences}
S.~Popa, \emph{Correspondences}, INCREST preprint, 1986.
\bibitem{powers:factors}
R.~Powers, \emph{Representations of uniformly hyperfinite algebras and their
associated rings}, Ann. Math. \textbf{86} (1967), 138--171.
\bibitem{powersstormer}
R.T. Powers and E.~St{\o}rmer, \emph{Free states of the canonical
anticommutation relations}, Commun. math. Phys. \textbf{16} (1970), 1--31.
\bibitem{radulescu:subfact}
F.~R{\u a}dulescu, \emph{Random matrices, amalgamated free products and
subfactors of the von {Neumann} algebra of a free group, of noninteger
index}, Invent. math. \textbf{115} (1994), 347--389.
\bibitem{radulescu:gamma}
\bysame, \emph{On the {$\Gamma$}-invariant form of the {Berezin} quantization
of the upper half plane}, Preprint, 1995.
\bibitem{radulescu:hecke}
\bysame, \emph{Arithmetic {Hecke} operators as completely positive maps}, C. R.
Acad. Sci. Paris \textbf{322, {S\'erie} {I}} (1996), 541--546.
\bibitem{renault}
J.~Renault, \emph{A groupoid approach to {$C^*$}-algebras}, Lecture notes in
mathematics, Springer-Verlag, 1980.
\bibitem{shlyakht:quasifree:big}
D.~Shlyakhtenko, \emph{Free quasi-free states}, Pacific J. Math \textbf{177}
(1997), 329--368.
\bibitem{stratila:modular}
{\c S}.~Str{\u a}til{\u a}, \emph{Modular theory in operator algebras}, Editura
Academiei, Bucharest and Abacus Press, England, 1981.
\bibitem{stratilazsido}
{\c S}.~Str{\u a}til{\u a} and L.~Zsid{\'o}, \emph{Lectures on von {Neumann}
algebras}, Editura Academiei, Bucharest and Abacus Press, England, 1975,1979.
\bibitem{dvv:free}
D.-V. Voiculescu, \emph{Symmetries of some reduced free product
{$C^*$}-algebras}, Operator Algebras and Their Connections with Topology and
Ergodic Theory, Lecture Notes in Mathematics, vol. 1132, Springer Verlag,
1985, pp.~556--588.
\bibitem{DVV:circular}
\bysame, \emph{Circular and semicircular systems and free product factors},
Operator Algebras, Unitary Representations, Enveloping Algebras, and
Invariant Theory, Progress in Mathematics, vol.~92, Birkh\"auser, Boston,
1990, pp.~45--60.
\bibitem{dvv:entropy3}
\bysame, \emph{The analogues of entropy and of {Fisher}'s information measure
in free probability theory, {III}}, Geometric and Functional Analysis
\textbf{6} (1996), 172--199.
\bibitem{dvv:book}
D.-V. Voiculescu, K.~Dykema, and A.~Nica, \emph{Free random variables}, CRM
monograph series, vol.~1, American Mathematical Society, 1992.
\bibitem{weinstein:ncgeom}
A.~Weinstein, \emph{Noncommutative geometry and geometric quantization},
Symplectic geometry and mathematical physics: actes du colloque en l'honneur
de {Jean-Marie Souriau} (P.~Donato et~al, ed.), Birkh{\"a}user, 1991,
pp.~446--461.
\bibitem{weinstein:irrat}
\bysame, \emph{Symplectic groupoids, geometric quantization, and irrational
rotation algebras}, Symplectic geometry, groupoids, and integrable systems
(P.~Dazord and A.~Weinstein, eds.), S{\'e}minaire {sud Rhodanien} de
G{\'e}om{\'e}trie {\`a} {Berkeley} (1989), Springer-Verlag, 1991,
pp.~281--290.
\bibitem{weinstein:KMS}
\bysame, \emph{The modular automorphism group of a {Poisson} manifold},
Preprint, 1996.
\bibitem{weinstein:poisson:notes}
A.~Weinstein, Ana~Cannas de~Silva, and Kevin Hartshorn, \emph{Lecture notes for
{Mathematics 277}, {Geometric} models for noncommutative algebras}, Preprint,
Berkeley, see {{\tt /$\wwwtilde$alanw/}}, 1997.
\bibitem{weinstein:xu}
Alan Weinstein and Ping Xu, \emph{Extensions of symplectic groupoids and
quantization}, J. Reine Angew. Math. \textbf{417} (1991), 159--189.
\end{thebibliography}
\end{document}