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\begin{document}
\title[Morita equivalence]{
Morita equivalence in algebra and geometry
}
\author{Ralf Meyer}
\email{rmeyer@math.berkeley.edu}
\thanks{This article has been prepared for the Spring 1997 Math 277 course at
the University of California at Berkeley, taught by Alan Weinstein}
\begin{abstract}
We study the notion of Morita equivalence in various categories. We start
with Morita equivalence and Morita duality in pure algebra. Then we consider
strong Morita equivalence for \CstarAlgebra{}s and Morita equivalence for
\WstarAlgebra{}s. Finally, we look at the corresponding notions for
groupoids (with structure) and Poisson manifolds.
\end{abstract}
\maketitle
\section{Algebraic Morita Equivalence}
The main idea of Morita equivalence in pure algebra can be illustrated by the
following example. Let~$R$ be any ring with unit, let~$\Mat(R)$ be the ring of
\PrM{n\times n}matrices over~$R$ for some $n\in\N$. If~$V$ is a (left)
\PrMn{R}module, then~$V^n$ is a \PrM{\Mat(R)}module in a canonical way
(matrix-vector multiplication), and the correspondence $V\mapsto V^n$ is
functorial. Conversely, every \PrM{\Mat(R)}module can be so obtained from some
\PrMn{R}module. Thus the rings $R$ and~$\Mat(R)$ have equivalent categories of
left modules.
\begin{definition}
We write~$\Mod{R}$ for the category of left \PrMn{R}modules. Two unital
rings are called \emph{Morita equivalent} if they have equivalent categories
of left modules.
\end{definition}
There is also a useful theory of Morita equivalence for rings with a ``set of
local units'', i.e.\ sufficiently many idempotents (cf.~\cite{Abrams:83}), but
things become far more complicated. Unless we have useful topologies around,
as in the case of \CstarAlgebra{}s, we assume all our rings to be unital.
Let $R$ and~$S$ be rings with unit. There is a standard way to get a functor
from $\Mod{R}$ to~$\Mod{S}$: If~$_SQ_R$ is any \PrM{(S,R)}bimodule and~$V$ is
an $R$-module, then $_SQ_R \otimes_R V$ carries a natural $S$-module structure.
Thus every \PrM{(S,R)}bimodule induces a functor from~$\Mod{R}$ to~$\Mod{S}$.
Taking the tensor product of bimodules corresponds to the composition of these
functors. Conversely, under some hypotheses, every (covariant) functor must be
of this form:
\begin{theorem}[Watts \cite{Watts:60}]
\label{the:EilenbergWatts}
Let~$T$ be a right-exact covariant functor from~$\Mod{R}$ to~$\Mod{S}$ which
commutes with direct sums. Then there is an \PrM{(S,R)}bimodule~$Q$ such
that the functors $T$ and $Q \otimes_R \blank$ are naturally equivalent.
Moreover, $Q$ is unique up to isomorphism of bimodules.
\end{theorem}
This result was discovered simultaneously by Eilenberg, Gabriel, and Watts
around 1960. As usual in homological algebra, the proof is trivial. Notice
that every equivalence of categories has to preserve direct sums and exact
sequences and thus satisfies the hypotheses of
Theorem~\ref{the:EilenbergWatts}. Hence we obtain
\begin{corollary} \label{cor:MoritaBimodules}
Two rings $R$ and~$S$ are Morita equivalent if and only if there are
bimodules $_RP_S$ and~$_SQ_R$ such that $_RP_S \otimes_S {}_SQ_R\cong
{}_RR_R$ and $_SQ_R \otimes_R {}_RP_S\cong {}_SS_S$ as bimodules.
\end{corollary}
This result implies that Morita equivalent rings also have equivalent
categories of right modules and bimodules. It is also easy to see that they
have equivalent lattices of ideals, so that the properties of being Noetherian,
Artinian, or simple are Morita invariant (cf.~\cite{Cohn:68}). They have
isomorphic categories of projective modules and thus equivalent K-theories.
More generally, a decent (co)homology theory should be Morita invariant, and
this is indeed true for cyclic homology, Hochschild homology (for
\PrMn{k}algebras) (cf.~\cite{McCarthy:88}).
%
% and also for the kk-theory developed recently by Joachim Cuntz.
%
Moreover, Morita equivalent rings have isomorphic centers. This implies that
Morita equivalent Abelian rings are already isomorphic. Thus Morita
equivalence is essentially a non-commutative phenomenon. This gives another
reason why so many homology functors are Morita invariant: Usually, they arise
as extensions of functors defined on a category of commutative algebras to a
category of non-commutative algebras. But Morita invariance imposes no
restrictions whatsoever on functors defined on a category of commutative
algebras, so that we can hope for a Morita invariant extension. Examples show
that if a functor can be extended ``naturally'', then the extension tends to be
indeed Morita invariant.
An important problem is to find conditions when two rings are equivalent.
Notice that we do not have to find two bimodules $P$ and~$Q$ because one of
them determines the other. In general, if the bimodules $P$ and~$Q$ implement
a Morita equivalence between $R$ and~$S$, we have
$$
Q\cong \Hom_S(P,S)\cong \Hom_R(P,R), \qquad
P\cong \Hom_S(Q,S)\cong \Hom_R(Q,R).
$$
This means that $Q$ and~$P$ are in some sense dual to each other. In the
purely algebraic setting, there is no natural way to turn an
\PrM{(S,R)}bimodule into an \PrM{(R,S)}bimodule; the nearest we can get is the
above relation between $P$ and~$Q$. For \CstarAlgebra{}s or groupoids, we
\emph{can} turn left actions into right actions using the adjoint operation or
inversion, which will slightly simplify matters there.
We let $\End(Q)$ be the ring of endomorphisms of the additive group~$Q$,
i.e.~$Q$ without the \PrM{(S,R)}bimodule structure. Then the right/left
operations of $R$ and $S$ on~$Q$ induce injective homomorphisms $R\to\End(Q)$
and $S\to\End(Q)$. The bimodule property asserts that the images of~$R$
and~$S$ in $\End(Q)$ commute, and in order to have a Morita equivalence, they
must be the full commutants of each other, i.e.\ $R'=S$, $S'=R$. This is clear
because $Q\otimes_R V$ is an \PrM{R'}module for all $V\in\Mod{R}$, and if our
inducing process gives all \PrMn{S}modules, we need $R'=S$. Thus~$R$ and the
right module structure of~$Q$ determine~$S$ as the commutant of~$R$ in
$\End(Q)$. Of course, not every module~$Q_R$ induces a Morita equivalence
(e.g.\ the zero module does not work).
\begin{theorem}[Morita \cite{Morita:58}, \cite{Morita:65}]
\label{the:ProGenMorita}
Let~$R$ be a ring with unit and~$Q$ a right $R$-module. Then~$Q$ induces a
Morita equivalence between~$R$ and $R'\subset\End(Q)$ if and only if~$Q_R$ is
a finitely generated projective generator.\footnote{An object~$X$ in an
Abelian category is a generator iff every object is a quotient of a direct
sum of copies of~$X$.}
\end{theorem}
Of course, the idea of ``representation equivalence'' is older than Morita's
work. His main contribution was to make formal definitions and to put the
various uses of this idea into a general theory. Besides the notion of
equivalence, Morita also studied a corresponding duality. Formally, this
consists of replacing covariant functors by contravariant functors. Since we
have $V^{\ast\ast}\neq V$ for an infinite-dimensional vector space, duality can
only hold if one restricts attention to finitely generated modules and assumes
that the underlying ring is Noetherian. Under these assumptions, the theory
goes through smoothly and yields:
\begin{definition}[Morita \cite{Morita:58}]
Let $\ModF{R}$ be the category of finitely generated left $R$-modules. If
$R$ and~$S$ are unital Noetherian rings, a duality is a pair $T\colon
\ModF{R}\to\ModF{S}$, $U\colon \ModF{S}\to\ModF{R}$ of contravariant
equivalence functors.
\end{definition}
\begin{theorem}[Morita \cite{Morita:58}]
Let $R$ and~$S$ be Noetherian rings. If there is a duality $(T,U)$ between
$\ModF{R}$ and~$\ModF{S}$, then there exists a bimodule~$_SQ_R$ such that
$T\cong \Hom_R(\blank,Q)$, $U\cong \Hom_S(\blank,Q)$. Moreover, the maps
$R,S\to\End(Q)$ are injective, and $R'=S$, $S'=R$.
\end{theorem}
Morita also has a necessary and sufficient condition for~$Q$ to induce a Morita
duality.
\section{Morita equivalence for \CstarAlgebra{}s and \WstarAlgebra{}s}
In these categories, we have considerably more structure and therefore restrict
our categories of modules.
\begin{definition}[Rieffel \cite{Rieffel:74b}]
A \emph{Hermitian module} over a \CstarAlgebra{}~$A$ is the Hilbert
space~$\HilS$ of a non-degenerate \PrMn{\ast}representation $\pi\colon
A\to\Bound(\HilS)$, together with the action $a\cdot\xi=\pi(a)\xi$ for $a\in
A$, $\xi\in\HilS$. If~$A$ is even a \WstarAlgebra{}, we assume in addition
that~$\pi$ is a normal map%
\footnote{In a \WstarAlgebra{}, every bounded increasing net of positive
elements has a least upper bound. A positive map $f\colon A\to B$ between
\WstarAlgebra{}s is called \emph{normal} if, for any bounded increasing
net~$(p_j)$ of positive elements of~$A$ with least upper bound~$p_\infty$,
$f(p_\infty)$ is the least upper bound of the net~$\bigl(f(p_j)\bigr)$.}%
%
and call~$\HilS$ a \emph{normal \PrMn{A}module}. In both cases, morphisms
are the intertwining operators, i.e.\ \PrMn{A}module homomorphisms in the
usual algebraic sense.
We call two \CstarAlgebra{}s \emph{Morita equivalent} if they have equivalent
categories of Hermitian modules and if the equivalence functors~$T$ are
\PrMn{\ast}functors, i.e.\ if $f\colon V_1\to V_2$ is a morphism, then
$T(f^\ast)=(Tf)^\ast$. Similarly, we call two \WstarAlgebra{}s \emph{Morita
equivalent} if they have equivalent categories of normal modules and if the
equivalence is implemented by \PrMn{\ast}functors.
\end{definition}
The category of Hermitian modules over a \CstarAlgebra{}~$A$ is equivalent to
the category of normal modules over the enveloping von Neumann algebra~$n(A)$.
Hence Morita equivalence of \CstarAlgebra{}s is really a von Neumann algebra
concept and too weak for most applications. We will soon define the more
restrictive concept of strong Morita equivalence for \CstarAlgebra{}s. As in
the purely algebraic case, we need more concrete criteria in terms of bimodules
for two algebras to be equivalent. Since we have to transport the Hilbert
space inner products, we need to put more structure on our bimodules:
\begin{definition}[Paschke \cite{Paschke:73}, Rieffel \cite{Rieffel:74a}]
Let~$B$ be a \CstarAlgebra{}. A \emph{pre-Hilbert \PrMn{B}module} is a right
\PrMn{B}module~$X$ (with a compatible \PrMn{\C}vector space structure),
equipped with a conjugate-bilinear map (linear in the second variable)
$\5{\blank}{\blank}_B\colon X\times X\to B$ satisfying
\begin{enumerate}
\item $\5{x}{x}_B\ge0$ for all $x\in X$;
\item $\5{x}{x}_B=0$ only if $x=0$;
\item $\5{x}{y}_B=\5{y}{x}_B^\ast$ for all $x,y\in X$;
\item $\5{x}{y\cdot b}_B=\5{x}{y}_B\cdot b$ for all $x,y\in X$, $b\in B$.
\end{enumerate}
The map $\5{\blank}{\blank}_B$ is called a \emph{\PrMn{B}valued inner product
on~$X$}.
\end{definition}
It can be shown that $\|x\|=\|\5{x}{x}_B\|^{1/2}$ defines a norm on~$X$.
If~$X$ is complete with respect to this norm, it is called a \emph{Hilbert
\PrMn{B}module}. If not, all the structure can be extended to its completion
to turn it into a Hilbert \PrMn{B}module. Actually, in Paschke's paper, the
inner product is linear in the first variable; and in Rieffel's paper, this
object is called a right (pre-)\PrM{B}rigged space.
This contains enough structure to transport Hilbert space inner products:
If~$V$ is a Hermitian \PrMn{B}module and~$X$ is a Hilbert \PrMn{B}module, we
can equip the algebraic tensor product $X\otimes_B V$ with an inner product
$$
\5{x\otimes v}{x'\otimes v'} = \5{\5{x'}{x}_B v}{v'}_V,
$$
where $\5{\blank}{\blank}_V$ is the inner product on~$V$. It can be shown that
this is non-negative definite and thus defines a pre-inner product on
$X\otimes_B V$. Thus factoring out the vectors of lenth zero and completing
gives a new Hilbert space $X\clotimes_B V$. This construction is functorial:
If $f\colon V_1\to V_2$ is a morphism of Hermitian \PrMn{B}modules, then
$\ID\otimes f$ extends to a bounded map $X\clotimes_B V_1\to X\clotimes_B V_2$.
If $e\in\lin(X,X)$ is a bounded operator commuting with the action of~$B$ by
right multiplication, then $e\otimes \ID[V]$ extends to a bounded operator on
$X\clotimes_B V$. However, the commutant of~$B$, in general, is not a
\CstarAlgebra{} because bounded operators may fail to have an \emph{adjoint}.
If $T\in\lin(X,X)$, an operator $T^\ast\in\lin(X,X)$ is called an adjoint
for~$T$ if $\5{Tx}{y}=\5{x}{T^\ast y}$ for all $x,y\in X$. Let~$E$ be the
algebra of all \emph{adjointable} operators on~$X$, i.e.\ operators that have
an adjoint. It is easy to see that an adjointable operator is necessarily
bounded and commutes with the action of~$B$. Moreover, $E$~with the natural
norm becomes a \CstarAlgebra{}. It is easy to see that $X\clotimes_B V$ is a
Hermitian \PrMn{E}module as expected. Moreover, the mappings $X\clotimes_B
V_1\to X\clotimes_B V_2$ induced by \PrMn{B}module homomorphisms are
\PrMn{E}module homomorphisms as desired, so that~$X$ induces a functor from
\PrM{B} to \PrMn{E}modules.
Let $B_0\subset B$ be the closed linear span of $\5{X}{X}_B=\{ \5{x}{y}, x,y\in
X\}$. If~$B_0$ acts trivially on~$V$, then $X\clotimes_B V$ is the zero
module, so that the functor induced by~$X$ fails to be faithful. Similarly,
the algebra~$E$ may be too big.
This can easily be seen from the example $B=\C$, $X=\HilS$
infinite-dimensional. In this case, $E=\Bound(\HilS)$ is the algebra of all
bounded operators on~$\HilS$. But~$\Bound(\HilS)$ has the non-trivial ideal
$\Comp(\HilS)$ of compact operators. It is well-known that $\Comp(\HilS)$ is
Morita equivalent to~$\C$: This means that every irreducible representation
of~$\Comp(\HilS)$ is a (possibly infinite) direct sum of copies of the standard
representation. But $\Bound(\HilS)$, as a \CstarAlgebra{} has more complicated
representations coming from the Calkin algebra $\Bound(\HilS)/\Comp(\HilS)$.
Here we have to be careful: As a \WstarAlgebra{}, $\Bound(\HilS)$ is Morita
equivalent to~$\C$, but not if we view it as a \CstarAlgebra{}.
This example suggests to look for an analogue of the ideal of compact operators
for Hilbert modules. The right approach is to let~$E_0$ be the closed linear
span of the ``rank one operators'' $\5{x}{y}_E\in E$ given by $\5{x}{y}_E z=
x\5{y}{z}_B$ for $x,y,z\in X$. It is easily seen that~$E_0$ is an ideal
in~$E$. Moreover, now the roles of $E_0$ and~$B_0$ are symmetric: We have just
defined an \PrMn{E_0}valued inner product on~$X$ and~$X$ is an \PrMn{E_0}module
by definition, only that we have exchanged left and right.
\begin{definition}[Rieffel \cite{Rieffel:74a}, \cite{Rieffel:76}]
Let $E$ and~$B$ be \CstarAlgebra{}s. By an \emph{\PrMn{E}\PrM{B}equivalence
bimodule} we mean an \PrM{E,B}bimodule which is equipped with \PrM{E} and
\PrMn{B}valued inner products with respect to which~$X$ is a right Hilbert
\PrMn{B}module and a left Hilbert \PrMn{E}module such that
\begin{enumerate}
\item $\5{x}{y}_E z = x\5{y}{z}_B$ for all $x,y,z\in X$;
\item $\5{X}{X}_B$ spans a dense subset of~$B$ and $\5{X}{X}_E$ spans a dense
subset of~$E$.
\end{enumerate}
We call $E$ and~$B$ \emph{strongly Morita equivalent} if there is an
\PrMn{E}\PrM{B}equivalence bimodule.
\end{definition}
If~$X$ is an \PrMn{E}\PrM{B}equivalence bimodule, it is easy to endow the
conjugate space~$\tilde{X}$, which is~$X$ as a set with the same addition and
scalar multiplication $\lambda\tilde{x}= (\conj{\lambda}x)\tilde{}$, with the
structure of a \PrMn{B}\PrM{E}equivalence bimodule. For example,
$\tilde{x}e=(e^\ast x)\tilde{}$. Moreover, it is not difficult to see that
strong Morita equivalence is an equivalence relation.
\begin{theorem}[Rieffel \cite{Rieffel:74a}]
Let~$X$ be an \PrM{E}\PrM{B}equivalence bimodule. Then $X\otimes_B\blank$
induces an equivalence between the category of Hermitian \PrMn{B}modules and
the category of Hermitian \PrMn{E}modules, the inverse being given by
$\tilde{X} \otimes_E\blank$. This functor preserves weak containment and
direct integrals.
\end{theorem}
The reason for Rieffel to introduce strong Morita equivalence was to improve
the understanding of induced representations of (locally compact) groups.
Let~$G$ be a l.c.\ group and let~$H$ be a closed subgroup. Then unitary
representations of~$H$ ``induce'' representations of~$G$. Moreover, the
representations of~$G$ obtained by this process are precisely those that admit
a ``system of imprimitivity''. In more modern language, the representations
that can be obtained by inducing from~$H$ are the covariant representations of
$(\BCI(G/H),G)$, where $\BCI(G/H)$ are the functions on~$G/H$ vanishing at
infinity and the action of~$G$ on $\BCI(G/H)$ is obtained from the left
translation action of~$G$ on~$G/H$. These results are due to Mackey (for the
separable case), but his proofs were based on rather unintuitive measure
theoretic arguments. In~\cite{Rieffel:74a}, Rieffel gave a new proof by
showing that the group algebra $C^\ast(H)$ is strongly Morita equivalent to the
crossed product $\BCI(G/H)\crossprod G$. Actually, he worked with the dense
subalgebras~$\BCC(\blank)$ of functions of compact support and showed that
$\BCC(G)$ can be given the structure of a pre-Hilbert \PrM{\BCC(H)}module.
Then he identified the algebra of ``finite rank operators'' on~$\BCC(G)$ with a
dense subalgebra of the crossed product $\BCI(G/H)\crossprod G$.
But there are also other, more non-commutative applications. For example,
if~$G$ is a compact group acting on a \CstarAlgebra{}~$A$ by automorphisms,
$\alpha\colon G\to\Aut(A)$, we can define a ``conditional expectation''
$p\colon A\to A^\alpha$, where~$A^\alpha$ is the fixed point algebra, by
averaging $p(a) = \int_G \alpha_x(a)\,dx$ with respect to Haar measure~$dx$.
Then $\5{a}{b}_{A^\alpha}=p(a^\ast b)$ turns~$A$ into a pre-Hilbert
\PrMn{A^\alpha}module. It can be shown that this gives us a strong Morita
equivalence of~$A^\alpha$ with a certain ideal of the crossed product algebra
$A\crossprod_\alpha G$. In the commutative case, this ideal is the whole
crossed product algebra, if and only if the action of~$G$ is free. Actually,
the case where~$G$ is not compact is very important but also much more subtle
(cf.~\cite{Rieffel:88}).
Another more elementary example is the following: Let $p\in A$, then the
corresponding left ideal~$Ap$ can be made into an
\PrMn{\cl{ApA}}\PrM{\cl{pAp}}equivalence bimodule with inner products
$\5{x}{y}_{ApA}=xy^\ast$, $\5{x}{y}_{pAp}=x^\ast y$. Subalgebras of the
form~$\cl{pAp}$ are the prototype of hereditary subalgebras, and the
corresponding hereditary subalgebra is called full if $\cl{ApA}=A$. This
example is of considerable theoretical importance because every strong Morita
equivalence is of this form: If $A$ and~$B$ are strongly Morita equivalent
\CstarAlgebra{}s, there is a \CstarAlgebra{}~$C$ that contains both~$A$ and~$B$
as full hereditary subalgebras. Together with a result of Brown on hereditary
subalgebras in~\cite{Brown:77}, this gives the following remarkable theorem:
\begin{theorem}[Brown-Green-Rieffel \cite{Brown-Green-Rieffel:77}]
\label{the:MoritaStable}
Let $A$ and~$B$ be \CstarAlgebra{}s with a countable approximate identity
(e.g.\ separable or unital). Then they are strongly Morita equivalent if and
only if they are stably equivalent, i.e.\ $A\otimes \Comp \cong B\otimes
\Comp$, where~$\Comp$ is the algebra of compact operators on a separable
Hilbert space.
\end{theorem}
Thus stable equivalence, which is of considerable importance in K-theory, can
be viewed as a separable version of Morita equivalence. Moreover, since the
class of separable or unital algebras is already rather large, one can expect
that properties that are invariant under stable equivalence are also Morita
invariant. For example, Morita equivalent \CstarAlgebra{}s have isomorphic
lattices of ideals and the same K\PrMn{{}}, E\PrMn{{}}, and KK-theory.
In~\cite{Combes-Zettl:83}, it is shown how to induce traces between Morita
equivalent \CstarAlgebra{}s. In~\cite{Combes:84}, Morita equivalence for group
actions on \CstarAlgebra{}s is defined, and it is shown that equivalent group
actions give rise to Morita equivalent group \CstarAlgebra{}s and reduced group
\CstarAlgebra{}s.
Now let us briefly discuss the situation for von Neumann algebras. If~$M$ is a
von Neumann algebra, a further requirement for (``normal'') Hilbert
\PrMn{M}\PrM{N}bimodules is that the maps $m\mapsto \5{x}{my}_N$ be
\PrMn{\sigma}weakly continous for all $x,y\in X$. On the other hand, we can
weaken the requirements for an equivalence bimodule, replacing density by weak
density. That this is possible is illustrated by the example $\C$,
$\Bound(\HilS)$. With these changes, the analogue of the
Eilenberg-Gabriel-Watts theorem is again true:
\begin{theorem}[Rieffel \cite{Rieffel:74b}]
Let $M$ and~$N$ be \WstarAlgebra{}s. Then every normal equivalence bimodule
implements an equivalence between the categories of normal \PrM{M} and
\PrMn{N}modules by a \PrMn{\ast}functor. Conversely, every such equivalence
is implemented by some normal equivalence bimodule.
\end{theorem}
It is easy to see that Morita equivalent von Neumann algebras have isomorphic
centers and isomorphic lattices of weakly closed ideals. Moreover, if $M$
and~$N$ are Morita equivalent and if~$M$ is of type
$X\in\{\mathrm{I,II,III}\}$, then the same holds for~$N$, i.e.\ Morita
equivalence respects the type of a von Neumann algebra. For types I and III,
the classification up to Morita equivalence is very easy:
\begin{theorem}[Rieffel \cite{Rieffel:74b}]
Two \WstarAlgebra{}s of type~I are Morita equivalent if and only if they have
isomorphic centers. Two von Neumann algebras of type III on separable
Hilbert spaces are Morita equivalent if and only if they are isomorphic.
\end{theorem}
As pointed out to me by Dimitri Shlyakhtenko, two factors $M,N$ of type~II on
separable Hilbert spaces are equivalent if and only if they are stably
equivalent as von Neumann algebras, i.e.\ $M\otimes\Bound(\HilS)\cong
N\otimes\Bound(\HilS)$ (this tensor product is in the category of von Neumann
algebras and is defined to be the weak closure of the spatial tensor product).
Thus every type \PrMn{\mathrm{II}_1}factor is equivalent to a
\PrMn{\mathrm{II}_\infty}factor, and conversely. The idea of the proof is to
turn a \PrMn{M}\PrM{N}Hilbert bimodule for two \PrMn{\mathrm{II}_1}factors into
a genuine (pre-)Hilbert space using the trace on one of them. The actions
extend to the completion, and it turns out that $M$~and~$N$ are commutants of
one another. Hence we obtain a correspondence in the sense
of~\cite{Shlyakhtenko:97} and can apply the theory for those.
It should be remarked that Morita equivalence of von Neumann algebras is not an
important technical tool, but at most a convenient way of formulating some of
the known results. For example, a von Neumann algebra is of type~I iff it is
Morita equivalent to a commutative von Neumann algebra.
\section{Morita equivalence for topological and symplectic groupoids}
Now we look at geometric analogues of Morita equivalence, first for locally
compact topological groupoids. The bimodule version still makes sense:
\begin{definition} [Muhly-Renault-Williams \cite{Muhly-Renault-Williams:87}]
\label{def:groupoidEquiv}
Let~$G$ be a locally compact topological groupoid with unit space~$G^{(0)}$
and source and range maps $s$ and~$r$. A locally compact space~$X$ with a
continuous, open map $\rho\colon X\to G^{(0)}$, which we call the
\emph{momentum map} and an action $\mu\colon G \ast X\to X$, where $G\ast
X=\{ (g,x)\in G\times X \mid s(g) = \rho(x) \}$, is called a \emph{left
\PrMn{G}space} if
\begin{enumerate}
\item $\rho\bigl(\mu(g,x)\bigr)= r(g)$ for all $(g,x)\in G\ast X$;
\item $\mu\bigl(\epsilon\bigl(\rho(x)\bigr),x\bigr)=x$ for all $x\in X$; and
\item $\mu(g\cdot h,x) = \mu\bigl(g,\mu(h,x)\bigr)$ whenever $(g,h)\in G\ast
G$ and $(h,x)\in G\ast X$.
\end{enumerate}
We write $g\cdot x = gx = \mu(g,x)$. A \emph{right \PrMn{G}space} is defined
similarly.
The action is called \emph{free} if $(g,x)\in G\ast X$ and $g\cdot x=x$
implies $g\in G^{(0)}$, i.e.\ only units have fixed points.
The action is called \emph{proper} if the map $(\mu,\ID)\colon G\ast X\to
X\times X$ sending $(g,x)$ to $(g\cdot x,x)$ is proper.
If~$H$ is another groupoid and if~$X$ is at the same time a left
\PrMn{G}space and a right \PrMn{H}space with momentum maps $\rho\colon X\to
G^{(0)}$ and $\sigma\colon X\to H^{(0)}$, we call it a
\emph{\PrMn{G}\PrM{H}bimodule} if the actions commute, i.e.\
\begin{enumerate}
\item $\rho(x\cdot h)=\rho(x)$ for all $(x,h)\in X\ast H$ and similarly
$\sigma(g\cdot x)=\sigma(x)$ for all $(g,x)$ in $G\ast X$; and
\item $g\cdot (x\cdot h)= (g\cdot x)\cdot h)$ for all $(g,x)\in G\ast X$,
$(x,h)\in X\ast H$.
\end{enumerate}
We say that a \PrMn{G}\PrM{H}bimodule~$X$ is an \emph{equivalence bimodule}
if
\begin{enumerate}
\item it is free and proper both as a \PrM{G} and an \PrMn{H}space;
\item the momentum map $\rho\colon X\to G^{(0)}$ induces a bijection of $X/H$
to~$G^{(0)}$; and
\item the momentum map $\sigma\colon X\to H^{(0)}$ induces a bijection of
$G\backslash X$ to~$H^{(0)}$.
\end{enumerate}
We call $G$ and~$H$ \emph{Morita equivalent} if a \PrMn{G}\PrM{H}equivalence
bimodule exists.
\end{definition}
The orbit space for a proper groupoid action is always locally compact
Hausdorff, and the projection onto the orbit space is open. Thus for an
equivalence bimodule the bijections $X/H\cong G^{(0)}$, $G\backslash X\cong
H^{(0)}$ are automatically homeomorphisms.
The action of a groupoid on itself by left and right multiplication turns it
into a \PrMn{G}\PrM{G}equivalence bimodule, so that Morita equivalence is a
reflexive relation. It is easy to see that it is also symmetric and
transitive. For the latter one uses the analogue $X\ast_H Y$ of the bimodule
tensor product: If $X$ and~$Y$ are \PrMn{G}\PrM{H} and \PrMn{H}\PrM{K}bimodules
respectively, then $X\ast Y = \{ (x,y)\in X\times Y \mid \sigma_X(x)=\rho_Y(y)
\}$, where $\sigma_X\colon X\to H^{(0)}$ and $\rho_Y\colon Y\to H^{(0)}$ are
the momentum maps. In order to get $X\ast_H Y$, identify $(x\cdot h,y)\sim (x,
h\cdot y)$ when this is defined. It is not difficult to endow this with the
structure of a locally compact \PrMn{G}\PrM{K}space and to see that this
process produces equivalence bimodules if $X$ and~$Y$ were equivalence
bimodules. Moreover, this tensor product is functorial (for ``equivariant''
continuous maps as morphisms).
\begin{corollary}
Let $G$ and~$H$ be Morita equivalent locally compact groupoids. Then the
categories of left (right) \PrM{G} and \PrMn{H}spaces are equivalent.
\end{corollary}
As in the algebraic case, under suitable hypotheses a left \PrMn{G}space
determines a groupoid~$H$ such that it becomes a \PrMn{G}\PrM{H}equivalence
bimodule \cite{Muhly-Renault-Williams:87}. To be more specific, let~$X$ be a
free proper \PrMn{G}space with a surjective momentum map~$\rho$. Let $X\ast X=
\{ (x,y)\in X\times X\mid \rho(x)=\rho(y)\}$. Then~$G$ acts freely and
properly on $X\ast X$ by the diagonal action $g(x,y)=(gx,gy)$. The orbit space
$H=G\backslash X\ast X$ can be endowed naturally with a groupoid structure over
$G\backslash X$ by putting $[x,y]\cdot [y,z]=[x,z]$, and this multiplication is
continuous. There is an obvious right action of~$H$ on~$X$ defined by $x\cdot
[x,y]=y$. It can be checked that this turns~$X$ into a
\PrMn{G}\PrM{H}equivalence bimodule. Moreover, if~$X$ was a
\PrMn{G}\PrM{H'}equivalence bimodule to start with, then we get $H\cong H'$.
There are many examples of Morita equivalent groupoids
\cite{Muhly-Renault-Williams:87}. If~$G$ is a transitive groupoid, $u\in
G^{(0)}$, then $r^{-1}(u)$ is an equivalence bimodule for~$G$ and the isotropy
group $r^{-1}(u)\cap s^{-1}(u)$ at~$u$, if $r$ and~$s$ are open maps. A
similar statement holds if $U\subset G^{(0)}$ is a subset meeting every
\PrMn{G}orbit. This applies especially to foliations (transverse submanifold
meeting every leaf). Moreover, we get that the groupoid associated to a
(Cartan) principal bundle (cf.~\cite{Mackenzie:87}) is equivalent to the
structure group of the bundle.
Another typical example is the following situation: Let $H$ and~$K$ be locally
compact groups acting freely and properly on a locally compact Hausdorff
space~$P$ such that the actions commute. Let~$H$ act on the left and~$K$ act
on the right. The commutativity assumption means that we get an action of~$K$
on $H\backslash P$ and an action of~$H$ on $P/K$. Then the space~$P$ is an
equivalence for the transformation groupoids $(H,P/K)$ and $(K,H\backslash P)$.
How is Morita equivalence of groupoids related to the algebraic notion?
Fix Haar systems $\lambda$ and~$\beta$ for $G$ and~$H$. Then we can form the
(full) groupoid \CstarAlgebra{}s $C^\ast(G,\lambda)$ and~$C^\ast(H,\beta)$ with
respect to these Haar systems. For the groupoids coming from the last example
above, it was already discovered by Green (cf.~\cite{Rieffel:80b}) that the
associated groupoid \CstarAlgebra{}s are Morita equivalent.
In~\cite{Muhly-Renault-Williams:87}, it is shown that this remains true in
general, with a proof similar to Rieffel's argument in~\cite{Rieffel:80b}:
\begin{theorem}[Muhly-Renault-Williams \cite{Muhly-Renault-Williams:87}]
\label{the:GroupoidMorita}
Let $G$ and~$H$ be locally compact, second countable, Hausdorff groupoids
with Haar systems $\lambda$ and~$\beta$. If there is a
\PrM{(G,H)}equivalence bimodule~$X$, then the (full) groupoid
\CstarAlgebra{}s $C^\ast(G,\lambda)$ and $C^\ast(H,\beta)$ are strongly
Morita equivalent.
\end{theorem}
The definition of the groupoid \CstarAlgebra{} depends on the choice of a Haar
system. However, the definition of a representation of a groupoid does not.
In the group case, Haar measure is essentially unique, but for groupoids, this
is no longer the case. Due to the correspondence of groupoid representations
and representations of the groupoid \CstarAlgebra{}, different choices of Haar
system certainly produce Morita equivalent \CstarAlgebra{}s. This still leaves
open whether we actually get isomorphic \CstarAlgebra{}s. At least in the case
of transitive groupoids, this is indeed the case:
\begin{theorem}[Muhly-Renault-Williams \cite{Muhly-Renault-Williams:87}]
Let~$G$ be a second countable, locally compact, transitive groupoid, let
$u\in G^{(0)}$, and let~$H$ be the isotropy group at~$u$. Let~$\lambda$ be a
Haar system for~$G$. Then there is a positive measure~$\mu$ on~$G^{(0)}$ of
full support such that $C^\ast(G,\lambda)$ is isomorphic to $C^\ast(H)\otimes
\Comp\bigl(L^2(G^{(0)},\mu)\bigr)$.
\end{theorem}
It is easy to see that $C^\ast(G,\lambda)$ must be strongly Morita equivalent
to $C^\ast(H)$. But the above refinement shows that we do not have to tensor
$C^\ast(G,\lambda)$ with the compact operators. This shows that the groupoid
algebra is stable and does not depend on the choice of Haar system.
By the way, it probably is not very interesting to look for criteria on
groupoids that are necessary and sufficient for the groupoid \CstarAlgebra{}s
to be Morita equivalent. This can already be seen by looking at groups. It is
easy to see that two groups are equivalent in the sense of
Definition~\ref{def:groupoidEquiv} iff they are isomorphic as topological
groups. However, if~$K$ is a compact group, then by the Peter-Weyl theorem its
groupoid \CstarAlgebra{} is a direct sum of copies of full matrix algebras, and
there are infinitely many such copies if and only if~$K$ is has infinitely many
elements. Thus any two infinite compact groups have strongly Morita
equivalent, even stably equivalent, group \CstarAlgebra{}s. However, there
seems to be no natural equivalence bimodule in this situation that can be
written down without knowing the full representation theory of the involved
groups.
If we drop all continuity assumptions, we get a notion of Morita equivalence
for algebraic groupoids without any further structure. More importantly, if
our groupoids carry additional differentiability structure, we should
strengthen our requirements on equivalences by asserting that the actions are
smooth in order to get an equivalence of the categories of smooth actions.
Moreover, the bijections of the orbit spaces $X/H$ with~$G^{(0)}$ and
$G\backslash X$ with~$H^{(0)}$ should be diffeomorphic. This follows if the
momentum maps are \emph{full}, i.e.\ surjective submersions.
\section{Morita equivalence for symplectic groupoids and Poisson manifolds}
\begin{definition}[Xu \cite{Xu:90}, \cite{Xu:89}]
Two symplectic groupoids $G$ and~$H$ with unit spaces $G^{(0)}$ and~$H^{(0)}$
are called \emph{Morita equivalent} if there are a symplectic manifold~$X$
and surjective submersions $\rho\colon X\to G^{(0)}$ and $\sigma\colon X\to
H^{(0)}$ such that
\begin{enumerate}
\item $G$~has a free, proper, symplectic \cite{Mikami-Weinstein:88} left
action on~$X$ with momentum map~$\rho$;
\item $H$~has a free, proper, symplectic right action on~$X$ with momentum
map~$\sigma$;
\item the two actions commute with each other;
\item $\rho$~induces a diffeomorphism $X/H\to G^{(0)}$;
\item $\sigma$~induces a diffeomorphism $G\backslash X\to H^{(0)}$;
\end{enumerate}
$(X;\rho;\sigma)$ is called an \emph{equivalence bimodule between $G$
and~$H$}.
\end{definition}
As expected, Morita equivalence is an equivalence relation among symplectic
groupoids. Since the notion is stronger than equivalence for topological
groupoids, Morita equivalent symplectic groupoids still have Morita equivalent
groupoid \CstarAlgebra{}s. Equivalence bimodules for symplectic groupoids were
also studied from a slightly different viewpoint and under the name of an
affinoid structure by Weinstein in~\cite{Weinstein:90}.
The point of introducing the stronger relation above is that we now get results
about the category of symplectic actions of our groupoids that are completely
analogous to the results for locally compact groupoids. In the proofs, it only
has to be checked that the symplectic structure can be transported. On the
other hand, topological problems almost disappear in this category. It is easy
to see that Morita equivalence of symplectic groupoids is an equivalence
relation.
\begin{theorem}[Xu \cite{Xu:90}, \cite{Xu:89}]
Let~$G$ be a symplectic groupoid over~$G^{(0)}$ and let $\rho\colon X\to
G^{(0)}$ be a full, symplectic, free, and proper left \PrMn{G}module.
Then~$G\backslash X$ is a Poisson manifold, and $H=G\backslash (X^- \ast_{G}
X)$ is a symplectic groupoid over~$G\backslash X^-$ in a natural way.
Moreover, $\sigma\colon X\to G\backslash X^-$ naturally becomes a symplectic
right \PrMn{H}module such that $(X;\rho;\sigma)$ is an equivalence bimodule
between $G$ and~$H$.
Conversely, if $(X;\rho;\sigma)$ is any equivalence bimodule between
symplectic groupoids $G$ and~$H$, then $H\cong G\backslash (X^-\ast_G X)$ as
symplectic groupoids.
\end{theorem}
As usual, if~$P$ is a Poisson manifold with bracket~$[,]$, then~$P^-$ denotes
the same manifold with bracket~$-[,]$.
Let~$G$ be a symplectic groupoid. We can consider the ``category'' of
symplectic left modules over~$G$, in which morphisms between symplectic modules
$F_1$ and~$F_2$ are Lagrangian submanifolds of $F_1 \ast_G F_2$ invariant under
the diagonal action of~$G$, and the composition of morphisms is the set
theoretic composition of relations. (This is not a true category since the
composition of two morphisms need not be a submanifold.) Then we obtain
\begin{theorem}[Xu \cite{Xu:90}, \cite{Xu:89}]
Morita equivalent symplectic groupoids have equivalent ``categories'' of
symplectic left modules.
\end{theorem}
One motivation for introducing Morita equivalence of symplectic groupoids is
the correspondence between integrable Poisson manifolds and \PrM{r}simply
connected%
\footnote{A groupoid is called \PrMn{r}\dots if all \PrMn{r}fibers have the
property \dots. Many authors call the range and source map, somewhat
unintuitively, $\alpha$ and~$\beta$ and thus write \PrMn{\alpha}\dots{}
instead.}
symplectic groupoids. It is possible to pull back the groupoid equivalence to
the base Poisson manifolds:
\begin{definition} [Xu \cite{Xu:90}, \cite{Xu:91}]
Two Poisson manifolds $P_1$ and~$P_2$ are \emph{Morita equivalent} if there
exists a symplectic manifold~$X$ together with complete Poisson morphisms
$\rho\colon X\to P_1$ and $\sigma\colon X\to P_2^-$ that form a full dual
pair with connected and simply connected fibers. Then~$X$ is called an
equivalence bimodule.
\end{definition}
The reason for requiring connected simply connected fibers is to exclude
certain cases that we do not want to be Morita equivalences. For example, with
this definition two connected symplectic manifolds are Morita equivalent iff
they have the same fundamental group. This somewhat complicated definition is
borne to make true the following theorem:
\begin{theorem} [Xu \cite{Xu:90} \cite{Xu:91}]
\label{the:PoissonGroupoidEquiv}
Let $P_1$ and~$P_2$ be integrable Poisson manifolds. Then $P_1$ and~$P_2$
are Morita equivalent if and only if their \PrMn{r}simply connected
symplectic groupoids are Morita equivalent.
\end{theorem}
Let~$G$ be an \PrMn{r}simply connected groupoid over~$P$. The main step
in the proof of Theorem~\ref{the:PoissonGroupoidEquiv} is to show that if~$X$
is a symplectic left \PrMn{G}module, then its momentum map $\rho\colon X\to
P$ is a complete symplectic realization and that, conversely, every complete
symplectic realization of~$P$ carries a natural left \PrMn{G}action. This also
proves the following theorem:
\begin{theorem}[Xu \cite{Xu:90}, \cite{Xu:91}]
\label{the:PoissonRealizations}
Equivalent integrable Poisson manifolds have equivalent ``categories'' of
complete symplectic realizations.
\end{theorem}
Furthermore, Theorem~\ref{the:PoissonGroupoidEquiv} immediately implies that
Morita equivalence is an equivalence relation among integrable Poisson
manifolds. This is not true for arbitrary Poisson manifolds: Already
reflexivity fails. Currently, it is not even known whether every Poisson
manifold has a \emph{complete} symplectic realization.
It is not difficult to see that an equivalence bimodule between two Poisson
manifolds induces a bijection between their leaf spaces. Moreover, Morita
equivalence takes into account the variation of the symplectic structures on
the leaves, which is measured by the fundamental class (cf. \cite{Xu:90},
theorem 1.2.5). This idea allows very precise statements about Morita
equivalence of regular Poisson manifolds. For example
\begin{theorem}[Xu \cite{Xu:90}, \cite{Xu:91}]
Let~$P$ be a regular Poisson manifold with symplectic fibration $\pi\colon
P\to Q$. Then~$P$ is Morita equivalent to~$Q$ with the zero Poisson
structure if and only if all the symplectic leaves of~$P$ are connected and
simply connected and the fundamental class vanishes.
\end{theorem}
\begin{theorem}[Xu \cite{Xu:90}, \cite{Xu:92}]
\label{the:SymplecticBundle}
Let $\pi\colon P\to M$ be a locally trivial bundle of connected, simply
connected symplectic manifolds. Then~$P$ is Morita equivalent to~$M$ with
zero Poisson structure.
\end{theorem}
Another interesting result is that Morita equivalent Poisson manifolds have the
same zeroth and first cohomology groups \cite{Ginzburg-Lu:92}. The example of
symplectic manifolds shows that no results about higher cohomologies can be
expected.
Theorem~\ref{the:PoissonRealizations} is an important tool for computing
symplectic realizations of Poisson manifolds, by reducing the problem to an
(apparently) simpler Morita equivalent manifold (see \cite{Xu:90},
\cite{Xu:92}). For example, Theorem~\ref{the:SymplecticBundle} reduces the
study of a locally trivial bundle of symplectic manifolds to that of symplectic
realizations of the base manifold with zero Poisson structure. Thus in order
to classify the complete symplectic realizations of a simply connected,
connected, symplectic manifold, we have to classify symplectic realizations of
a single point, which is rather easy. Careful bookkeeping shows that every
symplectic realization of~$S$ is of the form $\mathrm{pr}_S\colon S\times X\to
S$ for a symplectic manifold~$X$, where~$\mathrm{pr}_S$ is the canonical
projection. Another case that can be treated in this way is the orbit space of
a Hamiltonian action of a Lie group on a symplectic manifold, and the crossed
product of an integrable Poisson manifold with a Lie group acting on it
\cite{Xu:90}, \cite{Xu:92}.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}
\bibitem{Abrams:83}
Gene~D. Abrams, \emph{{M}orita equivalence for rings with local units},
Communications in Algebra \textbf{11} (1983), 801--837.
\bibitem{Brown:77}
Lawrence~G. Brown, \emph{Stable isomorphism of hereditary subalgebras of
{C}{$^\ast$}-algebras}, Pacific Journal of Mathematics \textbf{71} (1977),
335--348.
\bibitem{Brown-Green-Rieffel:77}
Lawrence~G. Brown, Philip Green, and Marc~A. Rieffel, \emph{Stable isomorphism
and strong {M}orita equivalence of {C}{$^\ast$}-algebras}, Pacific Journal of
Mathematics \textbf{71} (1977), 349--363.
\bibitem{Cohn:68}
Paul~Moritz Cohn, \emph{{M}orita equivalence and duality}, Queen Mary College
Mathematics Notes, Dillon's Q.M.C.\ Bookshop, London, 1968.
\bibitem{Combes:84}
Fran{\c c}ois Combes, \emph{Crossed products and {M}orita equivalence},
Proceedings of the London Mathematical Society \textbf{49} (1984), 289--306.
\bibitem{Combes-Zettl:83}
Fran{\c c}ois Combes and Heinrich Zettl, \emph{Order structures, traces, and
weights on {M}orita equivalent {C}{$^\ast$}-algebras}, {M}athematische
{A}nnalen \textbf{265} (1983), 67--81.
\bibitem{Ginzburg-Lu:92}
Viktor~L. Ginzburg and Jiang-Hua Lu, \emph{{P}oisson cohomology of
{M}orita-equivalent {P}oisson manifolds}, International Mathematics Research
Notices (1992), 199--205.
\bibitem{Mackenzie:87}
K.~Mackenzie, \emph{{L}ie groupoids and {L}ie algebroids in differential
geometry}, {L}ondon Mathematical Society Lecture Note Series, vol. 124,
Cambridge University Press, Cambridge, New York, 1987.
\bibitem{McCarthy:88}
Randy McCarthy, \emph{{M}orita equivalence and cyclic homology}, Les Comptes
Rendus de l'Acad{\'e}mie des Sciences. S{\'e}rie~1 \textbf{307} (1988),
211--215.
\bibitem{Mikami-Weinstein:88}
Kentaro Mikami and Alan Weinstein, \emph{Moments and reduction for symplectic
groupoids}, Kyoto University, Research Institute for Mathematical Sciences
Publications \textbf{24} (1988), 121--140.
\bibitem{Morita:58}
Kiiti Morita, \emph{Duality for modules and its applications to the theory of
rings with minimum condition}, Science Reports of the Tokyo Kyoiku Daigaku
(Tokyo University of Education). Section A \textbf{150} (1958), 83--142.
\bibitem{Morita:65}
Kiiti Morita, \emph{Adjoint pairs of functors and {F}robenius extensions},
Science Reports of the Tokyo Kyoiku Daigaku (Tokyo University of Education).
Section A \textbf{9} (1965), 40--71.
\bibitem{Muhly-Renault-Williams:87}
Paul~S. Muhly, Jean~N. Renault, and Dana~P. Williams, \emph{Equivalence and
isomorphism for groupoid {C}{$^\ast$}-algebras}, Journal of Operator Theory
\textbf{17} (1987), 3--22.
\bibitem{Paschke:73}
William~L. Paschke, \emph{Inner product modules over {B}{$^\ast$}-algebras},
Transactions of the American Mathematical Society \textbf{182} (1973),
443--468.
\bibitem{Rieffel:74a}
Marc~A. Rieffel, \emph{Induced representations of {C}{$^\ast$}-algebras},
Advances in Mathematics \textbf{13} (1974), 176--257.
\bibitem{Rieffel:74b}
\bysame, \emph{{M}orita equivalence for {C}{$^\ast$}-algebras and
{W}{$^\ast$}-algebras}, Journal of Pure and Applied Algebra \textbf{5}
(1974), 51--96.
\bibitem{Rieffel:76}
\bysame, \emph{Strong {M}orita equivalence of certain transformation group
{C}{$^\ast$}-algebras}, {M}athematische {A}nnalen \textbf{222} (1976), 7--22.
\bibitem{Rieffel:80b}
\bysame, \emph{Applications of strong {M}orita equivalence to transformation
group {C}{$^\ast$}-algebras}, Symposium in Pure Mathematics 1980. Operator
Algebras and Applications (Queens University, Kingston, Ontario) (Richard~V.
Kadison, ed.), Proceedings of Symposia in Pure Mathematics, vol.~38, 1982,
pp.~299--310.
\bibitem{Rieffel:88}
\bysame, \emph{Proper actions of groups on {C}{$^\ast$}-algebras}, Mappings of
Operator Algebras. Proceedings of the Japan-U.S.\ Joint Seminar, 1988
(University of Pennsylvania) (Huzihiro Araki and Richard~V. Kadison, eds.),
1990, pp.~141--181.
\bibitem{Shlyakhtenko:97}
Dimitri Shlyakhtenko, \emph{Von Neumann algebras and Poisson manifolds}, Survey
articles on geometric models for noncommutative algebras from Math 277,
Spring 1997, http://\discretionary{}{}{}math.berkeley.edu/\~{}alanw.
\bibitem{Watts:60}
Charles~E. Watts, \emph{Intrinsic characterizations of some additive functors},
Proceedings of the American Mathematical Society \textbf{11} (1960), 5--8.
\bibitem{Weinstein:90}
Alan Weinstein, \emph{Affine {P}oisson structures}, International Journal of
Mathematics \textbf{1} (1990), 343--360.
\bibitem{Xu:90}
Ping Xu, \emph{{M}orita equivalence of symplectic groupoids and {P}oisson
manifolds}, Ph.D. thesis, University of California at Berkeley, 1990.
\bibitem{Xu:91}
\bysame, \emph{{M}orita equivalence of {P}oisson manifolds}, Communications in
Mathematical Physics \textbf{142} (1991), 493--509.
\bibitem{Xu:89}
\bysame, \emph{{M}orita equivalent symplectic groupoids}, S{\'e}minaire
sud-rhodanien de g{\'e}om{\'e}trie 1989. Symplectic geometry, groupoids, and
integrable systems (MSRI, Berkeley), MSRI publications, vol.~20,
Springer-Verlag, 1991, pp.~291--311.
\bibitem{Xu:92}
\bysame, \emph{{M}orita eqivalence and symplectic realizations of {P}oisson
manifolds}, Annales Scientifiques de l'{\'E}cole Normale Sup{\'e}rieure,
4$^c$ s{\'e}rie \textbf{25} (1992), 307--333.
\end{thebibliography}
\end{document}