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Journal of Pure and Applied Algebra 5 (1974) 51-96. © North-Holland Publishing Companyi

MORITA EQUIVALENCE FOR C*-ALGEBRAS

AND W*-ALGEBRASMarc A. RIEFFEL

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University of California, Berkeley, Calif. 94720, U.S.A.

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Communicated by P. Freyd

Received 16 October 1973.

0. IntroductionA!

Two rings are said to be Morita equivalent if their categories of left modules areequivalent [26, 2, 6]. This provides a useful equivalence relation on rings which isconsiderably coarser than that of isomorphism. In this paper we study the corresponding notion of Morita equivalence for C*-algebras and W*-algebras, where as
their categories of left modules we take their categories of non-degenerate *-representations on Hilbert spaces (normal ones in the case of W*-algebras).
In the algebraic case, Morita’s basic theorem concerning Morita equivalence[26, 2, 6] gives a description of how two rings which are Morita equivalent are constructed from each other, namely that each must be the full endomorphism ring ofan appropriate type of module over the other. The main theorem of the present|

paper (Theorem 7.9) is an analogous description of how two C*-algebras or W *-algebrasiwhich are Morita equivalent are constructed from each other. Specifically, if M and N
are W* algebras, then we show that M and N are Morita equivalent if and only if each

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is the full algebra of “bounded” operators on a non-commutative analogue over theidother of the “C™-modules” which Kaplansky [21] defined over commutative
C*-algebras.
An important step in studying algebraic Morita equivalence is the study of functors between categories of modules which preserve certain categorical limits of thetype which any equivalence must preserve. The basic theorem concerning such func-{

tors is the Eilenberg—Watts theorem [11, 38, 2] asserting that any such functor isJ

equivalent to a functor consisting of taking tensor products with a bimodule. Similarly, a substantial portion of the present paper is devoted to studying functors be-:1

tween categories of modules over C*-algebras and W*-algebras and obtaining anb|analogue of the Eilenberg—Watts theorem (Theorem 5.5). This analogue states thatany such functor which is continuous in a certain sense is equivalent to a functor
consisting of forming a certain type of topological tensor product with one of the
non-commutative analogues mentioned above of Kaplansky’s “C*-modules”. This
topological tensor product is essentially just the inducing process which was studied

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52

M.A. Rieffel, Morita equivalenceiin [30] for C*-algebras, where it was shown that the formation of Mackey’s induced|

representations [23] is just a special case of this process, and that Mackey’s imprimi-

M.A. Rieffel, Morita equivalence

1. Modules over operator algebras
Let B be a C*-algebra (possiblyEework, and the results on functors obtained here will be useful in further study of induced representations.

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Since Morita equivalence is an equivalence relation on C*-algebras or W* algebras

53

without identit

clement).

Bya (lot) Hermitiantecar hr oss without ony lm)
By (e)Hermtar

ation 8 [8] with ne Toosponding action op we will noually use pnodule nota:tion w en working with
Hermitian modules.
andWare Hermitian B-modules,

then wel denote 0 Homg(¥, Ww) he Banach space (with the operator norm) of

which is considerably coarser than that of isomorphism, it will be a useful tool inaiistudying various aspects of these algebras, such as their; classification,although sucha study is not included here. The purpose of this paper is to lay the basic groundworkfor the theory. Much work remains to be done in obtaining detailed understandingof Morita equivalence for special classes of algebras.

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This paper is organized in the following way. Section 1 contains basic fects coniomitian ipa

cerning modules over C" -algebras and W™ -algebras. In particular, modules whic are
generators for the corresponding categories are studied, as these provide 2 tool whicha.
;sores ofo modules:
is important in later sections. In Section 2, catégories
over C . algebras
and W*-algebras are studied. In particular, the question of how much information
about an algebra can be recovered from its category of modules is considered, and

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tor
> oe cater 10 [251) Tots
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this question is seen to be closely related to Takesaki’s duality theorem " the repre:
sentation theory of C*-algebras [33, 3], as well as a number of other results in theliterature. General methods for constructing functors between categories of modules
over C*-algebras are described in Section 3. These methods generalize the inducing|||

process in [30]. Properties of these functors are studied in Section 4. In Section 5,
similar results for modules over W “algebras are considered, and the existence partof our analogue of the Eilenberg—Watts theorem is proved. Section 6 is devoted tothe uniqueness part of our analogue of the Eilenberg—Watts theorem. This involves
the self-dual modules introduced by Paschke [27] asa Ee
Tot © the noncommutative case of the “AW -modules” defined by Kaplansky [21] over commu-|||

sogether th the oy Hoe

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intertwining operators

By : iov : ot ars ee zee frie ho : race on on specific oe ort spaces.):
morphism [7] of ao the ech of all bounded, operators on which caries fhe idenae Teal ° le wo ; entation) By on ¢ oe ae“! i, ah in | 2] efiniHither . o en [ep © eons.) | ve the moaue we will mean theAs
wih pe ot
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Jo pe t Co presentation No zero: imeney" I" i" aoe i Ngatime
os 0 souee,repy ide oy ro “5 y X ca ot or oe Ne are y»wv (V, |tative 4 W"-algebras. Morita equivalence is studied in Section 7, and: the main theorem|

odule homomorphisms from F'to J This notation i compatible with at giver,ofpaperotherdescribing
oe porta
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are pufromthiseachis provedhowthere. "algebras
Finally, inwhich
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oll a category,oot gory, which
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C*-algebras and W*-algebras which are Morita equivalent are gathered together.

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I would like to thank George M. Bergman and John E. Roberts for some veryeC

" Ny or(8) be he envelons py

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alg

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helpful comments which are acknowledged more specifically later. I would also likehen n(B) es he property that he action of B on ay Hermitian eeto thank William L. Paschke for making available to me a pre-print of the paper [27]in which he introduced the self-dual modules which are used in Section 6.

518,15) ¢ o o @ a Cet radon
12.1.0]. Un,

Part of the research reported in this paper was carried out while I was visiting forheeft adjoint © the os

six weeks at the University of Marseilles. I would like to express my appreciation toDaniel Kastler and his colleagues for their warm hospitality during my visit. Anotherpart of this research was supported by National Science Foundation grant GP-30798X.

bauer on [19

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60]. tho form: gs ir N e,p.601],ianeror from the category of yr algebras to the

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To &
Be o ~) ni
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ished 2b ie
on betweer the
of A
fa
on in
O° es.
] ee ion br eves the ?
of Hormitinn H on jie o > ol ho o rtogory i” normal mE) modules. Thu he|to study Hermod-B, it suffices, as long as we are not considering this category to
carry additional structure, to study the category Normod-n(B). The converse is not

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54

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalencetrue however as there are many W *-algebras whose category of normal modules is| :not isomorphic to the category of Hermitian modules over any C"-algebra. For ex-W) is a group (under addition) and since the functor represented by U isclearly additive, it suffices to show that no non-zero element of Hom(V, W) is carried

ample, a category of Hermitian modules will always contain simple modules, that is,irreducible representations, whereas this need not be true for a category of normal
modules. For these reasons we will make many of our later definitions only for categories of normal modules. These definitions will then have immediate specializations
to categories of Hermitian modules.

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Let {V;} be a family of normal N-modules. Then the direct sum of this family,

to the zero map by this functor, that is, that given f € Homy(V, W) with f #0
there exists g € Homy (U, V) with f° g # 0. But, given such anf let V' be the “orthogonal complement of the kernel of £ Then V' is a non-zero submodule of V and fis
injective on V'. Since we are assuming condition (4) to hold, V' is isomorphic to a
direct sum of copies of non-zero submodules of U. Then f composed with the pro-|jection of U onto any of these submodules must be non-zero as desired.

denoted by © V;, is defined to be the Hilbert space direct sum of the V; together
with the obvious coordinate-wise action of NV. (To show that this action is well-defined it is necessary to use the fact that *-representations are norm-decreasing — see|

the comments after Theorem 2.11 of [29].) It is easily verified that © V; is a normal|

N-module. A bit of care must be taken in using direct sums of normal modules sincethese direct sums do not satisfy ‘the usual universal property for algebraic direct sums[25] if there is an infinite number of summands, but rather satisfy this universal
property only for families of homomorphisms whose family of operator norms issquare-summable. The definition of direct sums of Hermitian modules follows fromthat for normal modules. Categories of normal modules have the pleasant property

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that every submodule ofa normal module is a direct summand, the complementary|

We now investigate the subject of generators in categories of modules. These will
be an important tool later in our study of functors. In general category theory, an||

submodule being, of course, just its orthogonal complement.

iobject U is a generator [17, p.68] if the functor represented by U, namely!
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sufficient to take properties (3) or (4) below as the definition of a generator.
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1.1. Proposition. Let N be a W "-aigebra and let U € Normod-N.
conditions are equivalent:(1) U is a generator for Normod-N.

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(3) Every normal N-module is isomorphic to a submodule of a (possibly infinite)

1.2. Lemma on polar decomposition. Let C be a C*-algebra, let V and W be
Hermitian C-modules, and let f € Hom(V, W). Then f=p ) fl, where | f= (f* 1)?
(so | f1€ Homp(V, V)) and p is a partial isometry in Home (V, W) from the orthogonal complement of the kernel of | f| to the closure of the range of If.
Proof. Let X be the Hermitian C-module ¥ ® W, and define f€ Homp(X, X) by

fv, w)= (0, f(u)). Then it is easily verified that the usual polar decomposition for
operators ([32, Theorem 1.12] or {7, Appendix II1]) when applied to f yield thislemma. OJ

Since we are assuming (2) to hold, we can find a non-zero f € Homy (U,V). Thenfrom Lemma 1.2 applied to f there is & non-zero partial isometry pin Hom U,V).
Thus ¥ contains a non-zero submodule isomorphic to a submodule of U. If this sub-I

The four equivalent conditions of Proposition 1.1 are all stated in category-theoretic terms, and so apply immediately to the category of Hermitian modules over a|

C*-algebra. We now give a criterion for a module to be a generator in a category of|(4) Every normal N-module is a (possibly infinite) direct sum of copies of sub-|:to which we can apply (2) and the above argument. The result now follows by anapplication of Zorn’s lemma. 0:ii

direct sum of copies of U.
modules of U.

that form later.

module of Vis not all of V, then its orthogonal complement is a non-zero submodule]
Then the following(2) For any non-zero normal N-module V there is a non-zero element of
Homy (U, V).

It is clear that (3)-and (4) are equivalent. Finally, to show that (2) implies (4) weneed the following lemma, which we state for C*-algebras, since it will be useful in

We return to the proof of Proposition 1.1. Let V be a non-zero normal N-module.

V = Hom(U, V), is an embedding into the category of sets, that is, is injective on
spaces of morphisms. The next proposition gives other characterizations of generators
in categories of normal modules, and in fact for much of this paper it will be quite+

Homy(V,55

normal modules which is not category theoretic, and which is false for C*-algebras.

This criterion, which will be important later, follows readily from well-known facts

concerning von Neumann algebras, but these facts, unlike those used in the proof of
Proposition 1.1 above, are not entirely elementary.

Proof. We indicate why (1) implies (2). Let V be a non-zero normal N-module, so|that Hom, (V, V) contains non-zero elements. Since U is assumed to be a generator,
so that the functor represented by U is an embedding, it is easily seen to follow that~~.
Homy/(U, V) must contain non-zero elements as desired.
We show next that (4) implies (1). ket V and W be normal N-modules. Since

1.3. Proposition. Let N be a W*-algebra and let U be a normal N-module. Then U is|!

a generator for Normod-N if and only if the representation of N on U is faithful(that is, injective).
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M.A: Rigffel, Morita equivalence:

M.A. Rieffel, Morita equivalence

56

Proof. Suppose that U is a generator, and let Vy be the kernel of the corresponding‘

1.6. Proposition. Let B be a C*-algebra, and let U be any generator for Hermod-B.

normal homomorphism ofNV into the algebra of bounded operators on U. Then NV;
will be in the kernel of any representation on a submodule of a direct sum of copies|

Then the weak operator closure of the image of B as operators on U is isomorphic
to the enveloping W*-algebra n(B) of B.

of U, and so in the kernel of any representation of N on any member of Normod-/.
But any W *-algebra has a faithful normal module {32, Theorem 1.16.7], and so
Ny = {0}. Thus the representation of N on U is faithful.

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Proof. This is an immediate consequence of the above results and {8, 5.3.13]. ©

Conversely, suppose that the representation ofNV on U is faithful, and let

A number of well-known results concerning spatial isomorphisms of von Neumann

V € Normod-N. Then on examining [7, Theorem 3, p.53] we see that Vis obtained|

algebras can be given pleasant reformulation in terms of generators. For example,

by first taking an “ampliation”, that is, a direct sum of copies of U, then an “induction”, that is, a submodule of this direct sum, and then taking an isomorphism of|

the following is a reformulation of 7, ch. III §1, Theorem 3}:::

1.7. Proposition. Let N be a W *-algebra and let U and Uy be generators for Normod-/V.

this submodule with ¥. Thus condition (3) of Proposition 1.1 holds. And so thisIf each of U and Uy have both a separating and a cyclic vector, then U and Uy are iso-half of our proposition can be viewed as simply a reformulation of the indicatedmorphic."|

theorem in [7]. O
We now justify the comment made above that for C*-algebras the property of arepresentation being faithful is not categorical, by giving an example of two
C*-algebras which have isomorphic categories of Hermitian modules for which theisomorphism does not preserve faithfulness.

57

Among standard results in [7, ch. III] which can be reformulated in a similar way|

are Theorem 6 of § 1, Proposition 10 of § 6 and Corollary 7 of §8.
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‘
1.4. Example. Let 4 = cy, the C*-algebra of sequences of complex numbers whichconverge to zero, and let B be the C*-algebra of sequences {r, } of complex numbers:
2. Categories of modules over operator algebras:::
The main theme of this section is to discover how much information about a

the W*-algebra I_ of bounded sequences. In fact, it is easily seen that the dual of

Hermitian or normal modules. We do this not only because of the intrinsic interest -having the property that lim r,, = r,. Both algebras can be viewed as subalgebras of

C* algebra or W*-algebra can be recovered from knowing just its category ofeach algebra is /;, so that the double dual enveloping W*-algebra of each algebra isjust I. Thus the two algebras have isomorphic categories of Hermitian modules, each
But let ¥ be the subspace of /, consisting of sequences
isomorphic to Normod-/_.

of this question, but also because it leads to techniques which will be important inlater sections.
We recall from [2, p.56] that the center of a category is defined to be the collec-

having zero as first term. Then V is faithful as a B-module but not as an A-module.

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Actually, generators in the category of normal N-modules or Hermitian B-modules
have already played an important role in representation theory, although they havenot been called generators. For any C*-algebra B its “universal” representation ([8,2.7.6] or [32, 1.16.5]) is easily seen. to be a generator for Hermod-B, while for any
W*-algebra N its universal normal representation will be a generator for Normod-N.

|(In particular, generators always exist.) But other generators may be useful. For ex-|||tion of natural transformations from the identity functor on the category to itself.

The following proposition is an analogue of [2, Proposition 2.1, p. 56].
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2.1. Proposition. Let N be a W*-algebra. Then the center of Normod-N is an algebra
which is isomorphic with the center of N.
::

Proof. Any element c of the center of NV is easily seen to define a natural transforma-|tion #€ of the identity functor to itself by tS (v)=cvforveVe Normod-N. It isalso easily seen that the natural transformations form an algebra.
Conversely, suppose that ¢ is a natural transformation from the identity functor
to itself. Let U be a generator for Normod-V. Then by definition ty; € Hom (U, U),
and-also must commute with all the elements of Homy(U, U), that is, is in the center

alent. Any module quasi-equivalent to a generator is a generator. Thus the generators|

represented on U (Proposition 1.3), it follows from the von Neumann double com-form an equivalence class under the relation of quasi-equivalence.

:mutant theorem [7, p.41] that £7; corresponds to an element ¢ in the center of V. It

Proof. This is an immediate consequence of [8, 5.3.1(i)] together with Proposition 1.1||

is then clear that ¢ acts like ¢ on direct sums of copies of U and on submodules thereof,

ample, if B is a type I C*-algebra [8, 5.4], then the multiplicity free representation
quasi-equivalent [8, 5.3.2] to the universal representation of B will be a generator.
The enveloping W*-algebra of a C*-algebra is usually defined using the universal
representation, but the next two results show that any generator would do.

1.5. Proposition. Any two generators for Hermod-B (or Normod-NV) are quasi-equiv-

above. [J:!

|of Hom(U, U). Since we are assuming that Vis a W*-algebra and that:it is faithfully

and so on any element of Normod-N. O:

58

M.A. Rieffel, Morita equivalence:

M.A. Rieffel, Morita equivalence

2.2. Corollary. Let B be a C*-algebra. Then the center of Hermod-B is isomorphic tothe center of n(B).

|L

We can never actually recover a C*-algebra or W*-algebra just from its category ofmodules — in fact that is what makes Morita equivalence interesting. However, if
there is yet additional structure present, then it may be possible to recover the algebra.

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2.3. Corollary. Let N be a W*-algebra. Then N is a factor if and only if the center ofNormod-N is one-dimensional.

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For any W™*-algebraNV the category Normod-V has an additional piece of structure which will be of importance, namely the involution which assigns to eachf€Homy(V,W) its adjoint f* € Hom|(W, V). The following proposition is not

We now consider one example of this. For any W*-algebra N let H v (or just # when
there is no chance of confusion) denote the forgetful functor from Normod-V to thecategory of Hilbert spaces and bounded linear maps which assigns to every normal
N-module its underlying Hilbert space: The next theorem shows that we can recover
NN from the data consisting of Normod-V together with this forgetful functor. This

fact and its proof are just a slight generalization of a result (see Corollary 2.7 below)

surprising in view of the self-duality of Hilbert spaces.
i2.4. Proposition. Let N be a W*-algebra. Then the contravariant functor from
Normod-N to itself which is the identity on objects and carries each morphism f toits adjoint f* establishes a (conjugate linear) isomorphism of Normod-N with its
dual category.
.

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The proof of this proposition is trivial. The definition of a dual category can befound on [25, p.33].

discovered by John E. Roberts in the course of his work on mathematical physics,
where similar considerations are at play. (For a hint of this see the footnote on [9,p.217] and [10, Theorem 3.6].) I would like to thank him for showing me this resultand for several very enlightening conversations concerning the general subject.
2.6. Theorem. Let N be a W*-algebra, and let H be the forgetful functor from
Normod-N to the category of Hilbert spaces. Then the collection C of natural transformations from H to itself can be given in a natural way the structure of a W*-algebra,

and this W*-algebra is naturally isomorphic to N.

There is yet additional structure on Normod-V which will be of importance. We
have already mentioned the fact that each space Homp(V, W) is equipped with theoperator norm, with respect to which it is a Banach space. In addition, Homp(V, ¥)||!

Proof. Let £ € C. Then ¢ assigns to any ¥ € Normod-N a bounded linear operator tyon H(V). If scalar multiples, sums, products and adjoints of elements of C are defined

is not only a C*-algebra for the operator norm and involution, but is in fact a von|in terms of the corresponding operations on the associated operators, it is easily seen

Neumann algebra, and so can be equipped with the ultra-weak operator topology.
(Indeed, Hom, (V, W) can also be equipped with an ultra-weak operator topology,
as we will see in the next section.) In terms of this structure we can tell whether athat C becomes a *-algebra. A norm can be defined on C by
Iel=sup (It, 1: VE Normod-N}}
Vv)

W*-algebra is of type I, Il or III in terms of its category of normal modules.

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It is easily verified that this norm is finite, is a C*-algebra norm, and that Cis com-

2.5. Proposition. Let N be a W*-algebra. Then N is of pure type 1 (respectively typei|

plete for this norm, so that C becomes a C*-algebra. An analogue of the ultra-weak
operator topology can be defined by means of the linear functionals Pvuw defined

11, or type I) if and only if for every non-zero V € Normod-N the von Neumann|

for V € Normod-N, v,w €algebra Homp(V,V) is of pure type I (respectively type 11, or type III).

Py

Proof. From the definition of the type of a W* algebra [32, 2.2.9] it is easily seen|that any sub-W *-algebra of N will have the same type as N. But the image Ny, of N|

acting on ¥ will be a W*-subalgebra ofNV (as is easily seen from [32, 1.10.1, 1.16.2),
and so of the same typeas NV. But Homp(¥, V) is just the commutant of Ny, and:|so is also of the same type [32,2.9.6]. 0 .

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59

On the other hand, we shall see later (Corollary 8.13) that all type I, factors fordifferent values of n have equivalent categories of modules, so that they cannot be
distinguished by their categories. We shall also see that every type II factor has associated with it at least one type II; factor having an equivalent category of normal
modules, and conversely, so that one cannot tell whether a factor is of type II or
II; in terms of its category of normal modules.

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=

H(V) byty, w),-and it is easily verified that C becomes a W *-algebra for this topology.

For any n € NV we define a natural transformation #? from H to itself by("v=
4for v€ H(V),:V € Normod-N. It is easily verified that the correspondence n ~ #7 is

an isometric *-homomorphism of NV into C. We indicate now why this homomorphism
is surjective. Let £ €C, and let U be a generator for Normod-N. Then A is faithfully
represented on U (Proposition 1.3) and Hom (U, U) is just the commutant of N
acting on U. Now by the definition of a natural transformation #;; must commute
with every element of Hom (U, U). It follows from the von Neumann double commutant theorem [7, p.41] that there is an n € N such that n and ty coincide as
operators on U, It follows by arguments similar to those at the end of the proof of;60

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalenceLo
Proposition 2.1 that ty and n must coincide as operators ° n H( V) for every£
V’€Normod-N, that is, that # = #". Finally, it is easily verified that the analogue o
the ultra-weak operator topology on C defined above corresponds to the weak
topology ofN. OI
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:
)
.
CL.

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We notice from the above that the morphisms in Normod-V form a generalization|||of the commutant ofN, while the natural transformations of the forgetful functor

are like the double commutant ofNV.

||

2.7. Corollary (John E. Roberts). Let B be a ¢ algebra and let
H be the f orgetft ul|

functor from Hermod-B to the category of Hilbert spaces. Then the collection C ofnatural transformations fr on H to itself can be given in a natural way Ihe Structureof a W-algebra, and this W algebra is naturally isomorphic to the W*-enveloping
algebra n(B)

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of B.

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As T. Cartier pointed out to me, the above results are closely related tothe
Yoneda Lemma [25, p.61], but the Yoneda Lemma is not applicable since the forgetful functor is not representable.
}.
The above corollary is closely related to results of Takesaki [33] and Bichteler |[3], and in fact|

categories provide a more natural setting for their results. Specifically,

if

The above considerations are also ve ry closely related to the “bi“big group algebra iof Ernest [12, 13, 14], and to the enveloping dgebs of a covariant speters intro.
duced by Ernest in [15]. In all these papers the “options” employed can be reinterpreted to be natural transformations of a forgetful functor to itself.
As John E. Roberts also pointed out to me, the Tannaka duality theorem can bereinterpreted in a similar way, where the “Darstellungen der Dualhalbgruppe” of[36], or the “representations” of [5, Definition 2, p.196] or the “operations” on[20, p. 75] can all be viewed as natural transformations of the forgetful functor.

Undoubtedly the “operator fields” of [37] can be interpreted in a similar way, as-

can the maps J of [14, Remark 3.14]. Extension of these duality theorems to covari-ance algebras can be found in [16, Section 6]. Such generalizations of Tannaka’s

theorem seem to be more in the spirit of Tannaka’s original theorem than thosewhich involve Hopf algebras [22, 14, 34]. I also suspect that the above ideas are related to the work of Saavedra Rivano [31], but the situation here is not at all clear
to me.

Since non-isomorphic C*-algebras can have isomorphic enveloping W *-algebras(Example 1.4), one would not hope, in view of Corollary 2.7, to be able to recovera C-algebra from the data consisting of its category of Hermitian modules together
with the forgetful functor. However, one can imagine that if additional structure isadded (probably of a topological nature), then one could recover the C*-algebra itself. But it is not clear to me how to do this. Herewe will content ourselves with

Bis a C-algebra and if K is a Hilbert space, then the set Rep (B, K) of rep resenta-

reformulating in terms of categories the further results of Takesaki

tions of B on subspaces of K can be viewed as a category if the corresponding intertwining operators are adjoined. It is, of course, a full subcategory of Hermod-B. The
forgetful functor on Rep(B, K) has as its values on objects just subspaces of K. Then|bothe “admissible operator fields” of [33] and [3] are easily seen to correspond just|to natural transformations of this forgetful functor to itself. For example, it is {3,condition (iv), p.94] which corresponds to the requirement that a natural transformation must commute with morphisms. This is seen by using the following slight gener-|||

alization of [7, Proposition 3, p. 4];
2.8. Proposition. Let B be a C" algebra, and let We

61

Bichteler, inwhich they consider a topology on Rep (8, K). (It is not clear to me andhow to put asimilarly useful topology on Hermod-B.) ’Let B be a C*-algebra, and let K be an infinite-dimensional Hilbert space of dimen-sion large enough so that every cyclic representation of B canbe realizedon a subspace of K. Asin [33] and [3] we put on the objects of the category
Rep(B, K) a
topology, which in [18] is called the strong topology, but which we will call thestrong Fell topology. Specifically, any clement of Rep (3, K) whose underlying
Hilbert space is J < K can be put in correspondence with the homomorphism from

Hermod- B. Then any|

f &€ Homp(V, W)is a linear combination of two partial isometries in Homp(V, W).
.

Lo

|

.

:

Proof. According to Lemma 1.2 above, f=p £1, where p is a partial isometry mn
Homp(V, W) and I f1 is a positive element of Homp(V, "- But by [7, Proposition 3,p-4], If lis a linear combination of two unitary operators in Homg(V, V). O

|
|

oo
Co
Making the correspondence indicated above between admissible operator fields

and natural transformations of the forgetful functor, we see that [33, Theorem 2]
is just Corollary 2.7 above but with Hermod-B replaced by Rep (8, K). The proof

is almost the same except that XK may not be large enough to be the space of a gener:

ator for Hermod-B, and so one must work instead with a generating family of disjoint
elements of Rep(B, K). But this causes no difficulties.

B to the algebra of bounded operators on J which defines the corresponding representation of B. But every operator on J can be viewed as an operator on XK by de-fining it to be zero on the orthogonal complement of J. Thus the objects of Rep(B, K)
are in bijective correspondence with the *-homomorphisms of B into the algebra

L(K) of bounded operators on K which define (possibly degenerate) *-representationsof B on K. In this way the objects of Rep(B, K) correspond to certain functions
from B to L(K), and so if we equip L(K) with the strong operator topology, we canequip Rep (B, K) with the corresponding topology of pointwise convergence of functions. It is this topology which we call the stron g Fell topology. (As the lemma on||[3, p-90] shows, we could just as well have used the weak, ultra-weak or ultra-strong
operator topologies on L(K).)|

tion of the forgetful functor from Rep(B, K) to the ca tegory of Hilbert transformaspaces isi|

We would now like to define what we mean by saying that a natural

™
62

M.A. Rieffel, Morita equivalence:

continuous. For this purpose let Sub(K) denote the category whose objects consistof the subspace of K and whose morphisms consistof the bounded operators be-

tween these subspaces. The forgetful functor H is a functor from Rep(B, K) to
Sub(K), and a natural transformation from H to itself will be a map from the objects
of Rep(B, K) to morphisms in Sub(K). Thus we need a topology on the set of mor-+|

M.A: Rieffel, Morita équivalence:

3.1. Definition. Let B.be a C™-algebra. By a (right) pre-B-rigged space we mean avector space, X, over the complex numbers on which B acts by means of linear trans-|Pphisms of Sub(K). Now any morphism in Sub(X) can be viewed as a bounded operator on K by defining it to be zero on the orthogonal complement of its domain. We
obtain in this way a mapping from the set of morphisms of Sub(K) onto L(K), andwe equip the morphisms of Sub(K) with the pre-image under this mapping of thestrong operator topology on L(K). Note that the resulting topology on the morphisms

y
:
|
|

will not be Hausdorff. (As in the theorem on [3, p. 97] we could just as well use the63formations in such a way that X is a right B-module (in the algebraic sense), and on
which there is defined a B-valued pre-inner-product, that is, a B-valued sesquilinear
form (, )p conjugate linear in the first variable, such that

D(x, Xp >0forallx €X,(2) (x, »g)* =(y, x)g forall x,y EX,
and having the further property that(3) <x, yb)g =(x, y)gb forallx,y €X, b €B.

It is easily seen that if we factor a pre-B-rigged space by the subspace of the ele-

weak, ultra-weak or ultra-strong operator topologies on L(K).)

ments x for which (x, x) = 0, the quotient becomes in a natural way a pre-B-rigged

2.9. Definition. Let H be the forgetful functor from Rep(B, K) to Sub(K). A natural(x, x 35 > 0 for all non-zero x in X. On a pre-B-rigged space with definite innerspace having the additional property that its inner product is definite, that is,transformation from H to itself will be said to be continuous if it is continuous as afunction from the set of objects of Rep(B, K) equipped with the strong Fell topology| .
|to the set of morphisms of Sub(K) equipped with the topology defined above.

|(

The following is our reformulation of the main theorem of [33] and [3]:|i2.10. Theorem. Let B be a C algebra, and let K be a Hilbert space of infinite dimen-sion large enough so that every cyclic representation of B can be realized on somesubspace of K. Let H be the forgetful functor from Rep(B, K) to Sub(K), and let C
be the collection of natural transformations fromH to itself which are continuous in
the sense of the above definition. Then C forms a C*-algebra which is in a natural!

i

way isomorphic to B.

|;

This theorem can be proved by making trivial modifications of the proof given in£3], so we will not include a proof here.

3. Functors between categories of Hermitian modules:
Let A and B be C*-algebras. In this section we will study some general methods

|

|
:

|
;

for constructing functors from Hermod-B to Hermod-A.
One general method for constructing functors from Hermod-B to Hermod-A wasintroduced in [30], in terms of what were called there Hermitian B-rigged A-modules
these being a generalization to the non-commutative case of the “C*-modules” which
ee oo 1] introduced for commutative C*-algebras). For our present purposes
it will be useful to change slightly the definition of these objects (by no longer requiring the range of the B-valued inner product to span a dense submanifold of B,

and by requiring completeness).For this reason we will include the definitions here.
Contrary to the case in [30], there seems to be no advantage here in considering
pre-C-algebras, so we make our definitions only for C*-algebras.

|
|
i
|

product we can define a norm | Il by settinglx= lle,
xy. [12

G2)

x

A=

x, xp

for x € X. (See [30, Proposition 2.10] for the verification that this is indeed a norm.)

From now on we will always view a pre-B-rigged space with definite inner productas being equipped with this norm. Then the completion of X with respect to thisnorm is easily seen to become again a pre-B-rigged space. (This matter is discussed
around [27, 2.5].)

3.3. Definition. Let B be a C*-algebra. By a B-rigged space we will mean a pre-B-rigged
space, X, satisfying the following additional conditions:(1) If (x, x)5 = 0, thenx = 0, forall x €X,(2) X is complete for the norm defined in (3.2).

:

Viewing a B-rigged space as a generalization of an ordinary Hilbert space, we candefine what we mean by bounded operators on a B-rigged space, as was done in[30, Definition 2.3] and following [27, 2.5]. That these two definitions are equiv-

alent is shown in {27, Theorem 2.8]. For completeness we include here a definitionof bounded operators which lumps together the definitions from {30} and [27].

3.4. Definition. Let X be a B-rigged space. By a bounded operator on X we mean alinear operator, T, from X to itself which satisfies either of the equivalent conditions(1) for some constant ky we have|(Tx, Tx) Skp(x, x ’p
forallx ex;(1) Tis continuous with respect to the norm on X:{

and the condition|||(2) there is a linear operator, 7, on X, satisfying conditions (1) and (1) above,
such that(Tx, yy = (x, Ty )yforallx,y EX.
|

M.A. Rieffel, Morita equivalence| .

It is easily seen that any bounded operator on a B-rigged space X will automati-|

64cally commute with the action of B on X (because it has an adjoint).

We will denote by L(X) (or Lg(X)

serve weak containment of representations, as can be seen by arguments similar tointegrals when sense can be made of this.
We would now like to generalize the above construction to obtain a wider class ofbounded operators on X. Then it is easily verified that with the operator norm L(X)
is a C*-algebra ([30, Proposition 2.12] or comments after [27, 2.5]).

|

functors. As before, let A and B be C*-algebras, and now let n(B) be the W*-envel-

,
|
!
]

oping algebra of B. Let D-be any C *.subalgebra of n(B), and let X be a HermitianD-rigged A-module. Then we can use X to define a functor from Hermod-B to
Hermod-Aas follows. For any V € Hermod-B, we can view ¥ as a normal n(B)-moduleand we can then restrict the action to D, obtaining a (possibly degenerate) D-module.

3.5. Definition. Let 4 and B be C*-algebras. By a Hermitian B-rigged A-module wemean a B-rigged space, X, which is a left A-module by means of a x-homomorphismof 4 into L(X), and which is non-degenerate as an A-module in the sense that AX
is dense in X with respect to the norm on X.

B.

These inducing functors have some fairly nice properties. For example, they prethose in the proof of [30, Proposition 6.26], and they can be seen to preserve direct

if there is a chance of confusion) the set of all

This definition differs from [30, Definition 4.19] only in that we do not require
here that the range of the B-valued inner product on X span a dense submanifold of

65

M.A. Rieffel, Morita equivalence

We can then apply the construction described above using X to obtain an A-module

which will be non-degenerate for the reason mentioned at the end of the next tolast paragraph preceding 3.6. In this way we obtain a functor from Hermod-B to

Hermod-A. This process is slightly round-about, and we will see shortly that it can.

In [30, Section 5] it was shown how to use a Hermitian B-rigged A-module, X,
here.
to construct a functor from Hermod-B to Hermod-4. We recall the definition

If V € Hermod-B, then we can form the algebraic tensor product X @p ¥, and equip itwith an ordinary pre-inner-product which is defined on elementary tensors by(x 0, x 80 )= (x, X50 vy.
|||

be given a neater formulation. But first we will consider some examples.

|

3.7. Example. Let 4 = B = C([0, 1]), the algebra of continuous functions on theunit interval, The dual B' of B consists of the Borel measures on [0, 1], and so it is
ily of bounded Borel functions on [0, 1] can be viewed as a subrap=. Let D be the subalgebra of bounded functions on [0, 1] whichhave value zero except at a countable set of points, and let X =D. Let 4 and D act||

Completing the quotient of X* 8 V by the subspace of vectors of length zero, we
obtain an ordinary Hilbert space, on which 4 acts (by a(x © v)=ax ® v) togive as-representation of A. We will denote the corresponding Hermitian 4-module by|that for x,y €D the element (x, y);, viewed as a linear functional on B', is zero on
any purely continuous measure. Then it is easily seen that the functor determined

AY (or AV if there is a chance of confusion concerning which Hermitian B-rigged
A-module is being used). The only difference between the situation here and thatin [30] is that, because we no longer require the range of the B-valued inner producton X by pointwise multiplication, and define a D-valued inner product on X by)p, be the pointwise product of y with the complex conjugate of x. Noteletting (x,|by X as described above will assign to every Hermitian B-module its atomic part, that|is, the sum of the irreducible modules which it contains. A similar construction canbe carried out for any C*-algebra. This functor preserves neither weak containment

representation of B on V contains the range of the B-valued inner product, then indeed AV will be the zero-dimensional A-module. However, as long as we consider
the zero-dimensional A-module to be a non-degenerate A-module, we can still assert|

nor direct integrals.

easily seen that if the above construction is applied even to a degenerate B-module,|

we are assuming that X is non-degenerate. This remark will be useful shortly.
The above construction defines a functor if for ¥,W € Hermod-B and|on B to span a dense subset of B, we can no longer conclude that AY will not be thezero-dimensional A-module. In fact, it is easily seen that whenever the kernel of the

3.8. Example. Let 4 = B = C([0, 1]) as above, and let x = C(]0, 1]) with the evident

pointwise action of A. Let D be the subalgebra of B” = n(B) which is the range ofthat 4 V will be a non-degenerate A-module for all ¥ € Hermod-B. Actually, it is
it will nevertheless produce a non-degenerate 4-module, because of the fact thatf€ Homg(V, W) we define 4f € Hom, (AV,AW) on elementary tensors by

Af(x ®v) =x ® fv).
3.6. Definition. For any Hermitian B-module V the Hermitian A-module Ay will becalled the Hermitian A-module obtained by inducing V up to A via X. We will callthe corresponding functor the inducing functor determined by X.

|i|the map p of C([0, 1]) into B” which assigns to each f € C([0, 1]) the linear function

p(f) on B' defined by|p(f)(m)= | fdm,,

where m,, is the continuous part of the measure m. Let D, viewed as C([0, 1]), act
on X in the evident pointwise way, and define a D-valued inner product on X byxf(Feo =p(F2)8p=p(Jg) for fg eX.

56

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

Then it is easily seen that the functor determined by X assigns to every Hermitian
B-module its continuous part, that is, the complement of its atomic part. A similar

construction can be carried out for any C*-algebra.
—n=i

let D bethedrsof ounted Bore uncon on 0,1), vow
5s sobreed of 15) 5|

Then the corresponding functor will send all continuous modules to the zero module,
while it will permute the irreducible modules (and so the atomic modules) according|

oa mn partiouar, such 2 fanctor nly||

do cribed in Example 3.7. Let X be as in Example 3.7 with pointwise action of 4.

imilarsituation occurs in the313. Example. Let 4 = 8 = C([0, 1]). Notice that the D's of Examples 3.9 and 3.12

be viewed in a natural way as a subalgebra of n(B). Let X be the direct sum of theX’sfrom Examples 3.9 and 3.12. Then the corresponding functor carries each purely

paragraph before [30, Theorem 6.23]).

Horestian Dorigeed ule

continuous Hermitian module to an equivalent module, while it permutes the atomic

in the obvious way. Thenithe functor determined by Z doubles the atomic part of any Hermitian B-module

representations according to the permutation g of Example 3.12. In particular, this|functor is an equivalence of Hermod-B with itself which need not preserve weak con-while keeping the continuous part fixed (up to equivalence). This functor preservesiweak containment but notmeasurable field of irreducible repre-

are orthogonal subalgebras of n(B), so that the direct sum of these two algebras caniD Let YbeD with D-valued inner product defined in the usual way. It is easily
coon that the functor determined by Y is naturally equivalent to the identity functor
on [ermod® ! v molar o Se

67

anment or rect integrals and can carry a measurable field of modules to a non-t direct integrals.
direct integ)||

We will now reformulate the above general construction in a way which is morelegantbut which is somewhat more difficult to work with in specific examples suchelegpp
as those given above. Specifically, let 4 and B be C *.algebras, D a subalgebra ofdetermined by X is naturally equivalent to the identity functor. But this is also true|

if we let X = D. Thus non-isomorphic Hermitian rigged modules, in our present gener-

product X' ep, n(B) and define on it an n(B)-valued sesquilinear form by

|

ls
see

|

=
-

pr.

ey

the
eB

=
=
B of Example 1.4 and let D =n(B)=1_. LetX be
hy fy acting by pointwise multiplication on X, andwi th the obvious D - valued inner product on X. : Then it is easily seen that , the functor
.

EP .

.

eter. E] oles eeoon reewimsonos’,i,pahich are used
» pe a6 further conditions on the Hermitian rigged modules whichatection©).

(see

ons into irreducibles. .
In particular, this functor will preserve direct integrals, but it will not preserve wea||

except for the action of 4. Let g be a (possibly non-measurable) permutation ofi
X by settin
the points of [0, 1], and let 4 act on X by&(nf) (s)=hg(s) fs).

x

nn i)

=n*(x, x' pn’

This . is just a special case of the construction used in [30, Theorem 5.9] orin [27,Section 4], except that n(B) may be degenerate as a D-module. Nevertheless thesesquilinear form can be shown to be non-negative, either by imitating the proof
indicated for [30, Theorem 5.9] (and splitting the degenerate D-modules which

occur into their non-degenerate part plus null part) or by using [27, Proposition 6.1].
|

3.14. Proposition. Let A and B be C*-algebras, let D be a C “subalgebra of n(B) and
let X be a Hermitian D-rigged A-module. Let Y be the Hermitian n(B)-rigged
A-module obtained by equipping X ®p, n(B) with the pre-inner-product defined above,by factoring by the elements of length zero, and completing. Then the functor Fy
nitially defined on Hermod-n(B) but restricted to Normod-n(B) (which equals
nitia
Hermod-B) is naturally equivalent to the functor from Hermod-B to Hermod-A constructured from X by y the method described just before Example 3.7
7
pie
5.7.
Proof. Let ¥€ Hermod-B,

containment unless ¢ is continuous.
3.12. Example. Let 4 = B = C([0,1]), and let D and X be defined as in Exa mple 3.7,

(xen x'® "Vue) =x"-

3.11. Example. Let S and T be compact Hausdorff spaces. Let B= C (), A=C @.
and let b be the algebra of bounded Borel functions on > Bon]mutans “of mB) - 5 ” he use Define an action of Aon X bymapping g rom)r-E)=hg6) FE)i,:. A-module, and the
X becomes a HermitianD-rigged
fork €4, re Ls ES Then
X bl B-modules according to the mapping gq andcorresponding functor maps irreducible B-modu
; y
;
i
ibl
other modules according to their direct integral decompositin(B) and X a Hermitian D-rigged A-module. Then we can form the algebraic tensor:i”

and view ¥ as a normal #(B)module which can be re.

stricted to D. Then the natural map ¢}, from Y &,(5) V to X ep V defined on ele||mentary tensors byt((x 81) BV) =x 8 1|

is easily verified to provide the required natural unitary equivalence. OJ

-68

M.A. Rieffel, Morita equivalence|

In view of this result we may from now on restrict our attention to Hermitian|

M.A. Rieffel, Morita equivalencen(B)-rigged A-modules and the functors which they define from Hermod-B to

Hermod-4, though the earlier construction is still useful for considering specific

.

_

2
I=

VEN)

=F)

F(HI=IIF OIF

69

a
F=0r1%,
0

We now consider a somewhat less obvious property of the functors defined byexamples.
We remark that even by so restricting the rigged modules we consider to thosedefined over n(B) (in other words, by requiring the algebraD of the earlier construction to be n(BY itself); it is still possible for two non-isomorphic Hermitiann(B)-rigged A-modules to define equivalent functors, as is shown by Example 3.10.

|
|
:
|
|
|

Hermitian rigged modules. If ¥ € Hermod-B, then Homg(V, V) is not only a Banach
space, but in fact a von Neumann algebra, carrying in particular the ultra-weak
operator topology. More generally, given ¥V,W € Hermod-B, the space Hompg(V, W),
while not an algebra, still carries a natural analogue of the ultra-weak operator
topology, which, following the usage in [32] for W*-algebras, we will call simply
the weak topology. We will also have brief use for the analogue of the ultra-strong
operator topology on Homg(V, W).

4. Properties of functors||

With the eventual aim of characterizing the functors which are defined by
Hermitian n(B)-rigged A-modules, we now study the properties which these functors
possess. Throughout this paper we will always assume that the functors considered
are linear with respect to the linear structures on the Hom spaces.

|

4.4. Definition. Let V, W € Hermod-B. By the weak topology on Homg(V, W) we
will mean the topology defined by the linear functionals of the form
fir Sw, wo,"iiwhere {v;} and {w;} are sequences of elements from ¥ and W respectively such that>

As was mentioned earlier, Hermod-B carries a natural involution, that is, a con-3]012 < ooi’wil? <ooi:jugate linear contravariant functor of period two which is the identity on objects,|

namely the functor which takes each morphism to its adjoint.

|

4.1. Definition.
Let C and D be categories whose spaces of morphisms carry the
structure of complex vector spaces, and let C and D each have an involution, denoted!
||

defined by the seminorms of the form
Fro > 1) 121 no’i/by *. Then a linear functor F from C to D will be said to be a *-functor if|

where {v;} is a sequence of elements from V such that 3 | v; 12 <eo,F(f=F(f)"oofor every morphism fin C.

:

By the ultra-strong operator topology on Homp(V, W) we will mean the topology|

From [28, Theorem 1.4] it follows immediately that Hompg(V, W) isa dual|

Banach space, with the weak-# topology corresponding to the weak topology justdefined, in analogy with the well-known situation for von Neumann algebras.

4.2. Proposition. Let A and B be C*-algebras, let X be a Hermitian n(B)-rigged|

A-module, and let Fy be the corresponding functor from Hermod-B to Hermod-A(equipped with their natural involutions). Then Fy is a «functor.
The proof consists of a straightforward computation.
:|

principally [7, Lemma 2, p. 35):

We have the following analogue of well-known facts for von Neumann algebras,i- We now show that s-functors between categories of Hermitian modules are norm|

4.5. Proposition. Let V,W € Hermod-B. Then the ultra-strong operator topology on
Hompg(V, W) is stronger than the weak topology. The ultra-strongly continuous

linear functionals on Hompg(V, W) are exactly the linear functionals used in Defini-decreasing, in analogy with the well-known fact for s-homomorphisms between

tion 4.4 to define the weak topology. A net { fr} of elements of Hom z(V,W) con-C*-algebras.
4.3. Proposition. Let A and B be C algebras, and let F be a *-functor fromverges ultra-strongly to 0 if and only if the net { f f; } converges to 0 in the weak
topology of Homg(V, V).
The proof is obtained by making minor modifications to the proofs of the cor-

Hermod-B to Hermod-A.

responding facts for von Neumann algebras.

Then |

F(f) I< If I for every morphism f in Hermod-B.

Proof. Let f € Homg (VW) for V,W € Hermod-B. Now Homg(V, V') andHom 4(F(V), F(V)) are both von Neumann algebras, and F restricted to Hompg(V, V)
is a x-homomorphism, and so is norm decreasing {7, p. 8]. But"
so that|f€ Homg(V, V),|In view of Definition 4.4 it now makes sense to ask whether a mapping between
spaces of homomorphisms is normal, that is, continuous for the weak topology, andin particular we can ask this of the mappings between spaces of homomorphisms
defined by a functor.
:

r
70

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

4.6. Definition. Let 4 and B be C*-algebras, and let F be a #-functor from Hermod-B
to Hermod-A (or between full subcategories thereof). Then we will say that F'isnormal if for any V,W € Hermod-B (or its full subcategory) the mapping from
Homg(V, W) to Hom, (F(V), F(W)) defined by F is normal, that is, continuous for

|

71

|
|

4.9. Proposition. Let A and B be C*-algebras, and let F be a normal s-functor from
Hermod-B to Hermod-A4. Then F preserves (possibly infinite) Hilbert space direct
sums. That is, if {V;}is a family of elements of Hermod-B and if V =@® V,, then
F(V)=@ FV).

the weak topologies.
pole|

Proof. For each k let pj, denote the canonical mapping of © V; onto Vi. Then py

The main full subcategory which will interest us (in the next section) is the category of normal modules over a W *-algebra.

|

is the canonical inclusion of V into OV, Py Py is the identity map on Vg. Py Py
is the projection onto Pr (Vp); and p, p= 0if k #j. Then F(p;) F(py)" will be|F(pg) Fp)" =4.7. Proposition. Let F be a «functor from Hermod-B to Hermod-A. If for every|© F(V;) into F(® Vi). We must use the normality of F to show that this injectionis also surjective.

V € Hermod-B the homomorphism from Homg(V, V) to Hom, (F(V), F(V)) de-|

fined by F is normal, then F is normal.

|

Proof. Let ¥,W € Hermod-B. We must show that the map from Hompg(V, W) to

B

Actually, the next result shows that in order for a -functor to be normal it suf-

fices for it to be normal on the von Neumann algebras of form Hompg(V, V).

Hom 4(F(V), F(W)) defined by F is normal. We show first that it is continuous forthe identity on F(V), Fp)"

F(p;) will be a projection in F(® V;), andif k #7. It follows that the F(p,)* define a natural injection of

Now for any finite subset /V of the index set for the ¥}, the operator
py = 2pi py:

KEN}

is the projection of © V; onto the subspace corresponding to @® {Vi k EN}. Then
it is easily seen that the net {py } converges weakly to the identity operator on

the ultra-strong operator topologies. Let {fy } be a net of elements of Homg(V, W)

DV. Since we are assuming that F is normal, it follows that {F (ew)? converges

which converges ultra-strongly to 0. Then by the last part of Proposition 4.5 thenet { fy Ix }in Homg(V, V) converges weakly to 0. Since F is assumed normal on*.
Homg(V, V), it follows that the net { F(f, f,)}in Hom 4(F(4), F(A)) converges

weakly to the identity operator on F(©V;). From this it follows easily that the in-|iI

jection of 6 F(V;) into F(& V;) is surjective. 0!

that any functor preserves cokernels. Thus normal functors, in that they also pre-weakly to 0. But F is a #-functor, and so F(f; f) = F(/, 0) F(f,)- Then again by the

We remark that since in Hermod-B every short exact sequence splits, it follows

last part of Proposition 4.5 it follows that the net {F(f;)}in Hom, (F(V), F(W))
converges ultra-strongly to 0. Thus F is ultra-strongly continuous. But by using theserve direct sums, are a natural analogue of the right-continuous functors defined on

first two parts of Proposition 4.5 it is easily seen that any linear map between spaces[2, p. 58]. Indeed,.one of the main results of this paper will be an analogue forof homomorphisms which is ultra-strongly continuous is also weakly continuous. [Jnormal functors of the Eilenberg—Watts theorem for right-continuous functors([11,38] or [2, p.58)).

4.8. Theorem. Let A and B be C*-algebras, let X be a Hermitian n(B)-rigged A-module,
and let Fy be the corresponding functor from Hermod-B to Hermod-A. Then Fy isnormal.

x

We now give an example of a x-functor which is not normal,i

4.10. Example. We begin by considering a method for constructing functors from:p © Herinod|

the category, Hilbert, of Hilbert spaces (Hermitian C-modules) to itself. This consrction, wich oe Ll on
ves uegested to me A George M. oanhomomorphism
De be Pomalmp(V, V) V). Thenisthenormal.
Now let x ® fromv, x" Hom® v' € (Y:Fx (V)
given,
andx let f x omg(V,

|

L(K).
sy

Proof. According to Proposition 4.7 it suffices to show ha fig Hh

;

F

,

,

,

efine

ert,

,

=p")

.

:

:
and so it is clear that the composition of the homomorphism defined by Fy withthe linear functional defined by two elementary tensors is weakly continuous. This

»8/=D

|

will then also be true for any two finite tensors. But the oh tensors are Jone N
bia=efine a pre-inner-product(Fe(f) (x ov),x' ev") =(f ©), (x, x") pv),

Fy(V), and a routine argument using [7, Theorem 1, p. 38]

le

shows

defined by Fy is normal as desired. OJ

g

e map

:

K), andlet pon

Hom(K,

be anystate (normal or not)onHil
itself. Given
ilbert t itto itself.
Given V & Hilbert
ilbert,y|>and let F, »(V) be the corresponding Hilbert space obtained by factoring by the vectorsof fongth zero and oR

from this tha

Hom(K,We use > pp to constructfi
onetruc aahfunctorit F,, Su

from

Hom(X,reV) to Hom(X,Wy dotnet on nr ! RA
efinedby=ig

ho f for f €" en
Hom(X,

Then it is easily verified that 4 is continuous for the pre-inner-products defined
above, and so defines a continuous operator F »(1) from F, »(M) to F »(W). It is easily

We now examine an important property of normal s-functors.

|

seen that Fy, defined in this way, is a =-functor.

V).

[
72

M A. Rieffel, Morita equivalence| ;Now let K be an infinite-dimensional Hilbert space, and let L (XK) denote the|algebra of compact operators on K, so that L (K) is a two-sided ideal in L(K). Let p

is not normal, for it is easily seen that if # € Hom(¥V, W) and if / is compact, thenF(h) = 0. But the identity operator on any Hilbert space is the weak limit of com-73

Since it is normal as a functor from Hermod-V (Theorem 4.8), it follows immediatelythat its restriction to Normod-¥ is also a normal functor. OJ|

be a state of L(K) which is zero onL (C) (so that p is not a normal state). Then F, »M.A. Rieffel, Morita equivalence

Note that we will use the symbol Fy to denote both the functor from Hermod-N|

and its restriction to Normod-V, but this should not cause any confusion.

:|

We would like to prove conversely that every normal #-functor from Normod-N
to Normod-M is of the form Fy for some normal N-rigged M-module X. To do this
we must somehow produce such an X from any normal «functor. As motivation
for how to do this, we now consider how, given a normal N-rigged M-module X, we

pact operators.

5. Normal modules and the Eilenberg—Watts theorem

can recover X from Fy. Now for any ¥ € Normod-N and any x € X we can define

We have seen that if B is any C*-algebra, then Hermod-B is isomorphic to
Normod-n(B). Thus any functor from Hermod-B to Hermod-4 can equally well be||a bounded linear operator ty, from Vito Fy(V) = My byXn||

In general ty will not respect any actions of M or N. However, if W € Normod-V
and if f € Homy,(V, W), then it is easily seen thatty) =xev.

viewed as a functor from Normod-n(B) to Normod-#(4). We are thus led naturally
to make a general study of functors between categories of normal modules overW*-algebras.
Let

M and NV be W*.algebras. Since Normod-N is a full subcategory of Hermod-V,

we know what is meant by a normal #-functor from Normod-V to Normod-M

xview of the results of Section 3 we would expect to construct such functors in terms

_F

wf)

|

In

|

=

x

Xx (Ne pV)

for any v € V. This says that, for fixed x, the family of maps ty as V ranges overof some kind of M-N-bimodules with N-valued inner product. We will show now that
this is in fact the case, the main difference from the previous section being that wemust here ensure that the range of the functor consists of elements of Normod-M:||

Normod-/V forms a natural transformation from the forgetful functor Hy, from
Normod-V to Hilbert (the category of Hilbert spaces), to the forgetful functor A Mr
from Normod-M composed with F x» that is, a natural transformation from Hy, toand not just Hermod-M. In what follows we will, as before, refer to the ultra-weak|

Hp © Fy. Thus we can hope to recover X as the collection of all natural transforma-

operator topology on W*-algebras as just the weak topology.

tions from Hy to Hy, © Fy. We will see later (Section 6) that this can often enoughbe done, but not always, since we have already seen that non-isomorphic Xs can

5.1. Definition. Let M and N be W*-algebras.By a normal N-rigged M-module wemean a Hermitian N-rigged M-module, X, which has the added property that for|every x,y € X the linear map m = (x, my Jy from M to N is normal, that is, weakly
continuous.

define equivalent functors (Example 3.15 with 4 taken to be L(K)).
In view of the above considerations it is appropriate to develop some tools forhandling natural transformations. Note that the category Hilbert can be considered:||

to be Normod-C, and that the definition of a normal N-module says that the functor
Hy; is a normal functor. Similarly Hy, © Fy will be a normal #-functor. Thus it is5.2. Theorem. Let M and N be W™-algebras, and let X be a normal N-rigged M-module.

appropriate to study natural transformations between pairs of normal #-functors

For any V € Normod-N let Fy (V?) be the Hermitian M-module obtained by inducing
V up to M via X. Then Fy(V) is in fact in Normod-M. In this way we obtain a normal

from Normod-V to Normod-M, where in our first applications we will take M to be
C. But the following results will also be useful when we study the uniqueness of X

sfunctor Fy from Normod-N to Normod-M.

|

Proof. Consider two elementary tensors x ® v and x’ @ v' in F(V'). Then the func-|

5.3. Proposition. Let M and N be W *-algebras, and let F and G be normal functors|

from Normod-N to Normod-M. Let s and t be two natural transformations from F
to G. If sy; = ty; for some generator U for Normod-V, then s = £.

tionalme(mix ov), xo vy = (xmx " vv)

is weakly continuous because of the normality of X and V. But the elementarytensors span a dense subspace of Fy (V), and from this it is easily seen that Fy(V)is normal.

Thus the inducing functor from Hermod-V to Hermod-M defined by X as in

Section 3 carries the full subcategory Normod-V into the full subcategory Normod-M.

in the next section.

Proof. Let VV € Normod-V,||

and assume first that there is an isometric isomorphism

fof V onto a submodule of U, so that f* fis the identity operator on ¥. Then===

G(N)sy=sy

FN) =ty

A(f)=G(f)t, .

Multiplying on the left by G(f i) and using the fact that G(f ") G(f) is the identity

operator on G(V), we find that si, = 74.

pa

74

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

Suppose now that Vis an arbitrary element of Normod-N. Then V is the direct|

sum of copies of submodules of U (Proposition 1.1), on which we have just seen thats and ¢ agree. From the fact that normal s-functors preserve direct sums (Proposition
4.9) it follows easily that s and ¢ agree on V. [J

|

:

*

.

75

.

.
thermore fg g =f. Then a straightforward calculation using
these results showsat|t,= G(g") TF(g),so that £, does not depend on the choice off|

now¥,W € Normod-V
isometric
fandSuppose
phand that
W sessectively
oh ounand thatat thereor areome
i isomorphisms
somo iTm

which we will denote by c(V) (or ¢,(V) if there is a chance of confusion), is thecommutant of the action of N on V, and is a von Neumann algebra. As a result, V|Wwe" g gahave g(or more precisely Hy(V)) can also be viewed as a normal ¢(V)-module. Then if F|

is a normal #-functor from Normod-V to Normod-M,

F will define a normal homo-!

The

traightf

morphism from ¢(V) = Homy(V, V) into Hom (F(V), F(V)), and via this homomorphism F(V') also can be viewed as a normal ¢(V)-module.

|.
In preparation for the next result we note that if. ¥ € Normod-#, then Homy(¥, V),

5.4. Proposition. Let M andN be W “algebras, and let F and G be normal functors|(as well as normal Mmodules). Then the assignment 10 any natural pransformation N|||

ith the actions ve both M and c(V) as required

|

cea

op
> t;; is injective by Proposition
5.3. We must show that

Supposenow that F is a normal «-functor from Normod-N to Normod-M, , andPP

that we wish to find an X such that F is naturally equivalent to F ¢- In the discussion
:

Japa

.

on

ee i

|

fof V onto a submodule of U, so that f* fis the identity operator on V. Define ¢},j Benerator

nd He

M

alues

ind So plays litte role)
|

|

:

Ir

=

:

Ho

ba, foe Proposttio N on , Cut p
»

th

:

For Simplicity Hoy

M>3and then teldng Ee

ave values in bart
Nc(M\Y>

We.

somit we an.

Since T commutes with the action of M, it is clear that ty, € Hom, (F(V), G(V)).

.

oS. Theorem. oer id ond Ne

another isometric isomorphism of V into U. Then fg~:€ Homp(U, U) = ¢(NV). Since

|

|

and

one of the main theorems of this paper, namely, the analogue of the Eilenberg—Watts

theorem ([11, 38] or [2, p. 58]).

We must show that the definition of #}, does not depend on the choice of f. Let g be

TF(fg")=G(fg") T.

tha

identity of F(U). Thus r © ¢ agrees on U with the identity natural transformationfrom F to itself. : It follows from Proposition 5.3. that 7 © ¢ is the identity naturaltransformation. In a similar way it is seen that 7 © r is the identity natural transforma-aand assume first that there is an isometric isomorphism

T is assumed to commute with the actions of c(V) defined by F and G, it follows

thatusing the fact that normal #-functors commute with direct sums (Proposition 4.9).

"!

actions of both M and ¢(V). We wish to extend T to a natural transformation from

that

n

recedingANSE
Proposition 5.3fromit wasthe collection5.4 ofthisall iDlA, suggested
haplt oF. thatButwe in takeview as ofX Propodtion

Let T be a linear transformation from F(U) to G(U) which commutes with thet,=G(f*) TF(f)calculati

If Vis an arbitrary element of Normod-V, the definition of ty and the verificationthat it is a natural transformation now follow in a routine way by decomposing V
;
:
.
.
.
into the direct
sum of modules which
are isomorphic to submodules of U, and by

this mapping is surjective.

by

(V,s

tion from G to itself. Thus # is a natural equivalence as desired. LJ

Normod-¥, and so is a linear transformation which commutes with the action of M.
’
i
.
; the
t
fi
f
tural transformation, #; must also commute withF to G. Let V € Normod-V,

nv

T is invertible, theninversebe extended
oa Finally,a] iftransformation.
say 7, itsfrom
G 10 canF, and(rf)=byvothe 1 aboveol 9 processoe

Proof. If ¢ is a natural transformation from F to G, then #;; must be a morphism in.

ywhich is the characteristic property of natural transformations,ipiv Gt) which commute both with the genors orM and wih the os os ,oN)ponddefinedby F'and tr G respectively.
to invertible
rmations. Under this bijection natural equivalences cor
respondto invertible transfo

Furthermore, the mapping4n a straightiorward calculation showsty Fi) =Gh)t,,

from Fto Gof the linear transformation ty from F(U) to G(U) establishes ” vection
between the natural transformations from F to G and the linear transformations fromBut bY . hes on 0 Jo:

TF(ghf*) = G(ghf*) T.

|

from NNto Normod-M. Let Ube a generator for Normod-N, and let¢(N) = Homy/(U, U), so that A U) and G(U) can be viewed as normal o(N)-modules
=
NAY,
’
f
.

Pp u
y
Cot
omy/(U, U) = ¢(N), so tha

EA

hm id raomp/(U,I algebras, md he g ho ‘ norma

HL

U),

isan

-

+: wctor / on-

0 F x mn fact, ifps id generator for Normod-N, and if
then

an ta

X= Hom, (U, FU).

0

be

r

76

M.A. Rieffel, Morita equivalence|

Proof. Let U be a generator for Normod-/V and let X be defined as indicated. We mustfirst show in what way X is a normal N-rigged M-module. Now since the action of N
on U commutes with that of ¢(N), X becomes a right N-module if an action is definedby (xn) (u) =x(n(u)) forx €X, n €N, u € U. Since U is a generator for Normod-V,
N is faithfully represented on U (Proposition 1.3), and it follows from von Neumann’s
double commutant theorem [7, p.41] that NV can be identified with Hom,(U, U).
Now if x,y € X, thenx"y € Hom,(U, U), and so, from the above, x™ y can beviewed as an element of NV. Accordingly we define an N-valued inner product on X
by.
xyly=x"y

|
|
|
|
|
|

for x,y € X. It is easily verified that in this way X becomes an N-rigged space. (Thisis a special case of [30, Example 4.26].)
Now F(U) is an M-module, and so we can define an action of M on X by||(mx) W) = m(x(u))|

form eM, x € X, u € U.1t is easily verified that with this action X becomes a
Hermitian N-rigged M-module. Furthermore, for any x,y € X the map"

M.A. Rieffel, Morita equivalence77

6. Uniqueness and self-dual modules
We have seen that non-isomorphic rigged modules can define equivalent functors(Example 3.10). In this section we shall see that if we impose an additional condition on the rigged modules considered, then the correspondence between normal#-functors and rigged modules becomes essentially bijective. We will motivate thisadditional condition from a different direction. Let 4 and B be C -algebras, and let
X be a Hermitian n(B)-rigged A-module as in Section 3. Then X defines a functorfrom Hermod-B to Hermod-4, and so from Normod-(B) to Normod-n(4). But according to Theorem 5.5 this functor should be defined by a normal n(B)-rigged
n(4)-module. It is natural to ask whether the action of 4 on X can be extended to- an action of n(4), in which case X would become the desired normal n(B)-riggedn(A)module. We will see shortly that it is not always possible to so extend the action of A. But, in view of the universal property of n(4), one sufficient condition
for extending the action of 4 is that the algebra L(X) of all bounded operators on
X be a W*-algebra. The following example shows that in general L(X) is not a
W*-algebra even when X is defined over a W*-algebra.

|me (x, myly =x" myfrom M to NV is clearly continuous for the weak topologies on M andN. Thus X is in

6.1. Example. Let V =I", the W™-algebra of bounded sequences. Let cq be the;|fact a normal N-rigged M-module.

ideal of I consisting of sequences converging to zero, viewed as an N-rigged spacein the usual way. Let X = cj ® I, viewed as an N-rigged space in the obvious way.

|:

We can now use X to define a normal #-functor Fy from Normod-N to Normod-M
as in Theorem 5.2. We would like to show that Fis naturally equivalent to Fy.
Since F and F 'y are both normal s-functors, it suffices, in view of Proposition 5.4,ito show that there is an invertible transformation from Fy (U) to F(U) which commutes with the actions of bothM and ¢(V). Accordingly, define a bilinear map from|X X Uinto F(U) by (x, u) = x(t). This map is clearly N-balanced, and so defines a|

Two more simple calculations show that T commutes with the actions of both M
and ¢(V). Thus what remains to be shown is that T is surjective.
:
|
!

|
|

ready been found by Paschke [27], namely, that the N-rigged space should be selfdual in a sense analogous to that for ordinary Hilbert spaces. We will see that such

results here, stated in our terminology.
oo

.

Co

6.2. Definition. Let Bbea ¢ -algebra, and let X be a B-rigged space. Then X is said
to be self-dual if every continuous (for the usual norm on X ) B-module homomor-

phism R from X into B (with B acting on itself on the right) is of the formz in Hom, (5 (U, W+). We can view z as an element of X whose range is hu WL. Choose

R(x) =, x asu € U such that z(u) # 0. Then T(z ® u) = z(u) € WL, which contradicts the fact that
W is the range of 7. Thus 7 must be surjective. I

action of 4 on X does not extend to an action of n(4).
It is thus natural to look for conditions on an N-rigged space which will ensureself-dual spaces are also the ones which give uniqueness in our analogue of theEilenberg—Watts theorem. For the convenience of the reader we recall Paschke’s

Now it is clear that the representation of ¢(V) on U is faithful, and so according
to Proposition 1.3, U is also a generator for Normod-c(V). Let W denote the range ofWis not zero. Then it follows from Proposition 1.1 that there is a non-zero element

isnot a W -algebra. Furthermore, if 4 is the algebra of 2 X 2 matrices all of whose .
entries come from Cg» acting on X in the obvious way, then it is easily seen that thisthat its algebra of bounded operators is a W*-algebra. Now such conditions have al-linear map T from X ey U into F(U) satisfying T(x ® u) = x(u). A simple calculation
then shows that T is isometric, and so extends to an isometry of Fy (U) into F(U).

T in F(U), and view F(U) as a ¢(V)-module. Since T is a ¢(V) homomorphism, W is ac(N)-submodule of F(U), and so its orthogonal complement W* is also. Suppose that

Then it is easily seen that L(X) consists of the 2 X 2 matrices whose diagonal elements come from I= but whose off-diagonal elements come from Cg, and this algebra;|

forsomey€ X.

|

6.3. Proposition ([27, Proposition 3.10]). Let N be a W algebra and let X be aself-dual N-rigged space. Then L(X) is a W*-algebra. In fact, if N, denotes the pre-78

M.A. Rieffel, Morita equivalence

5

dual of N, then the functionals
Tr p((y,forpEN,, x,y €X,

M.A. Ri¢ffel, Morita equivalence

6.7.

Tx)
’
N
T € L(X), span a norm-dense submanifold of the predual of::Lemma (Gram—Schmidt process). Let M be a W*-algebra, and let Y be an M-rigged

space which has polar decompositions in the sense of Proposition 6.4. Let {x, ..., x}
be a finite family of elements of X. Then there is an orthonormal family { Vp oer Vikof elements of Y and a family {ml: 1 <i,j €k} of elements of M such that

LE).

x=
23 {y,ml: 1<j<Kk}

In addition, Paschke showed that any N-rigged space can be enlarged to form aself-dual N-rigged space [27, Theorem 3.2], the enlargement being in a vague sense
an analogue of the weak operator closure of an operator algebra. The developmentsin this section will as a by-product lead to a different (and perhaps simpler) proof||

be defined over a W*-algebra or for X to be self-dual, as can be seen by considering_|

an ideal in n(B).

y r 2) ={y ry 2M pry
Since the left-hand side is zero, it follows that

3

Paschke also indicated the importance of “polar decompositions” for the study

Lemma 1.2 — but see the comments after Proposition 6.12), but we state it here to|provide a backdrop for our use of “polar decompositions” in the following results.

|

range projection of |x|, that is, the smallest projection e in N such that e| x | = tx 1.

|

!

N

|

6.5. Theorem. Let C be a C algebra and let V and W be Hermitian C-modules. Let

N=Hom¢(V, V)and X = Hom(V, W), and letN act on the right on X by composition of operators. On X define an N-valued inner product by (x, Yiy = x*y forx,y €X. Then X is a self-dual N-rigged space.

Proof. It is easily seen that X is an N-rigged space. Furthermore, Lemma 1.1 shows(2) <¥;, y; 2

the proof continues in the same way to imitate the proof of the usual Gram—Schmidt

We now complete the proof of Theorem 6.5. Let R be a continuous N-modulehomomorphism of X into N. We wish to find an element y of X such that
R(x) ={yg, x)y for all x €X. We will do this by first finding yz, which will be an
V). We begin by defining YR on elements of W of the form

Zx{(v;), where {x;} and {v;} are finite families of elements of X and V. Specifically,
if we knew that yp existed, then we would have

Pp Qax,0)) = 23 (yp x) (0) =23 Rx), .
Accordingly, we define YR by|

We must show that yy is well defined. For this it clearly suffices to show that if

Yr ( xv)= bh R(x;) ; .

Zx;(v;) = 0, then 2 R(x;) vy; = 0. Now, since X has the polar decomposition property

by Lemma 1.1, we can apply Lemma 6.7 to find an orthonormal family {y;} of

as is seen immediately from calculating the B-inner-product of y — y (y,yp with

6.6. Definition. Let M be a W algebra, let Y be an M-rigged space, and let {y;} be

2M

|
i
|

|

elements of X and a family {n] } of elements of NV such that x; = = ¥; n} for each i.

itself. To continue the proof we shall need to take orthogonal expansions:a family of elements of Y. We will say that { ¥;}1is an orthonormal family of elementsof YifpYaPhy =p

by the comment before Definition 6.6. Thus y, and y, are orthogonal. The rest ofelement of Hom(W,a class of self-dual spaces which will be of great importance in

.

_

P=

process. [J

6.4. Proposition ([27, Proposition 3.11]). Let N be a W*-aigebra and let X be a self:
dual N-rigged space. Then each element x of X can be written uniquely in the form
ylx|, where |x 1=((x, x)y 2 andy is an element of X such that {V, Vy is the

Y is an element of a B-rigged space such that (y, y ) is a projection, then y=y(y, yp,_0=(rpy 2of self-dual rigged spaces. We shall not need the next result (except in the form ofthat X has polar decompositions in the sense of Proposition 6.4. We remark that if= 0. Let the polar decompositionof zy be yy my. Let p = (y,, ¥,)y so that p is the range projection of m,. Theneither the algebra of compact operators on a Hilbert space as a rigged space over itself, or cy as a rigged space over itself, or, more generally, any C*-algebra B which is.

and letXp

Then a routine calculation shows that (24, yr

We remark that in order for L(X) to be a W*-algebra, it is not necessary for X to

We will now exhibit
this section.

foreachi.

Proof. Let the polar decomposition of x, be ym,
2y=Xy = ypof this result (Proposition 6.10).

79

{
|

‘

1s a projection in M for each i.
iLetz=3y; R( DR Then a straightforward calculation, using the fact thatyj { Yj Yi N= yj for each j and that R is an N-module homomorphism, shows that_

R(x) ={z, x;y

:

|
80

\

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

for each i. Then, since Zx;(v;) = 0, we have

(6.8)

|

2IR(xe) v= 2042, x,y v= 2" (2x,(0)) = 0,||

normal family of elements of X, and {v;} is some family of elements of V. As above,yiy; (x) = 2 R(y)v; = 2|y;(v) .
.

Then|
I ysR (2 yziri.

2 yw).
[2|

But straightforward calculations show that|—*Iz¥12=1z*z 1 =12R(7,)(», yon ROY) * (|I=IROI<IRINzI=IRIIZ™
N,

so that 1z*

I< URI, andys|

IS URI.

*-_—

R(X) () =y R Ce) =<y rR XN ©,so that R(x) = (yg, x Jy as desired.

= Homp/(U, U), so that Hom, (U, F(U)) is a normal N-rigged M-module as shownin the proof of Theorem 5.5. Now, by Proposition 5.4, there is a bijection between
Y and Hom,(U, £(U)), and this bijection is easily seen to preserve the actions of
M and N on these two spaces. Using this bijection we can transfer the N-valued innerproduct on Hom, (U, F(U)) to Y, so that Y becomes a normal N-rigged M-module.
We show that this inner product on Y does not depend on the choice of generator.
Suppose that Uj is another generator for Normod-V, and assume at first that there.
.
:
is an isomorphism f of U onto a submodule of Uy. Because U and Uj are generators,
the maps =f" nfis an isomorphism of N acting on Uj, onto N acting on U. But for
5,t €7, itis easily seen from the properties of natural transformations that.

shy =p,ty, flso that 5, 7, and sy, ty, represent the same element of N. If Uj does not contain a||copy of U, then we can compare both U and Up with the generator U @ Uj, to obtainhe general case: Pinal, Sinee Homan oA) hy i” self-dual module by Theorem.5, it follows
that Y is a self-dual normal
N-rigged
M-module.
Suppose now that X is a self-dual normal N-rigged M-module, and that F xisliequivalent to F, so that Y can be viewed as the space of natural transformations

Finally, we must show that yp commutes with the action of C. Now XV and its
complement are both invariant under C, and so, since VR is zero on the complement,||

it suffices to show that yg commutes with the action of C on the submanifold X V.

from Hy, to Hy, © Fy. We show that the mapping of X into Y defined by assigningtox €X the natural transformation t* defined by t*(v) = Xo v (as in the discussion

after Theorem 5.2) isan isomorphism of X with Y. Now this mapping is easily seen

But this is shown by a simple calculation. It follows that yp itself commutes withto be an injective isometric M-N-bimodule homomorphism. The only issue is whether

the action of C. [J

it is surjective.

Let t € Y. We wish to find z € X such that t = #%. Let U be a generator for

We would now like to show that any two self-dual normal N-rigged M-modules|

which define equivalent functors from Normod-V to Normod-M are isomorphic(contrary to what happens if we drop the condition of self-duality).
:
6.9. Theorem. Let M and N be W*-algebras, and let F be a normal =-functor from
Normod- to Normod-M. Then the space Y of natural transformations from Hy to!

Hy © F has a natural structure as a self-dual normal N-rigged M-module. Furthermore,whenever convenient later.) Let U be a generator for Normod-V, and let ¢(V) =|We can now extend YR ‘by continuity to the closure of XV, and define Yr to be
zero on the orthogonal complement of XV. Then for any x € X and v € V we have—_—

(For simplicity of notation we omit the forgetful functors Hy, and Hy here, and|setz = 2; (R( A) I so that as in the derivation of (6.8) above we haveequivalence classes of normal x-functors from Normod-N to Normod-M and isomorphism classes of self-dual normal N-rigged M-modules.
Proof. The elements of Y form a vector space under pointwise operations. Furthermore, Y forms an M-N-bimodule with actions defined by
f
= mt
+
=
(m ly (v) = m( (©)( n)y, @)=t p(n).

as desired.
We now show that YR» defined so far only on the linear submanifold XV of W,
is bounded there. Note that y Rr is clearly linear. Now from the calculations above it
is clearly seen that any element of XV is of the form Z y;(v;), where {»;}is an ortho-

81|

Normod-, and let c(V) = Homy (0, U). Then for any x € X, (t*);; and ¢ y are bothin Hom, (U, Fxy(U)), so that ti (ty € Hom, (U, U), and so can be viewed as anelement of NV by the double commutant theorem. We thus have a map R from X into
N defined bysy
R(x)=t Uv ( dy :if X is a self-dual normal N-rigged M-module and Fy is equivalent to F, then X isnaturally isomorphic to Y. In particular, any two self-dual normal N-rigged M-modules|

It is easily verified that R is an N-module homomorphism (with NV acting on itself on|

which define equivalent functors are isomorphic. Thus there is a bijection between|

the right), and that R is bounded (by Il #7;1). Since X is assumed to be self-dual, it
follows that there is an element z of X such that R(x) = (z, x yy forall x € X, that is,£, (t %), = (zx).

,

iP

r
82

M A. Rieffel, Morita equivalence|

But for any x € X and 4,4’ € U, we then have(ty), xeu'y=(u, t,t), @))=(u, (z,x)yu)=(zouM.A. Rieffel, Morita equivalence|

x au’).

Proof. For

VV € Normod-/V define a natural transformation ¢ from Fz to Fy by|

It follows that ¢; = (#*);. From Proposition 5.3 it then follows that ¢ = #2.

tyzov)=zev

ors °z ve

We have thus shown that the correspondence from isomorphism classes of self-|

dual normal N-rigged M-modules to equivalence classes of normal s-functors is injec-|

tive. But this correspondence is also seen to be surjective by Theorem 5.5 if we note|

Itis easily verified that fy is an isometry. We must use the density

i. bo § on that v is surjective. If it is not, then there is aw € Fy (V), w #0,

i. bn °i) :that the X constructed there is self-dual by Theorem 6.5. [Jceto the range of ty. Since w #0, there must exist y €X,

|
:

vE V

at 4 N vu, w) #0. Then the functional hon X defined by (x) = (x & v, w)

ERbe v ro on X. Buh 0 easily seen to be in the pre-dual of X, and so Z
weak-xWe now indicate how the above results can be used to give a new proof of Paschke’s theorem [27, 3.2] that any N-rigged space can be embedded in a self-dual N-rigged

83

dense in

X,

[J

)
.
)
Tom he consicerations leading to Proposition 6.10 we also get a representation.

space. In fact, we show this for any normal N-rigged M-module (Paschke’s case being|that in which M = C). Let Z be a normal N-rigged M-module. Then we can form the:or self-dual

normalN-rigged M-modules:

functor F and apply Theorem 5.5 to form X = Hom, (U, Fz(U)), so that F zs
equivalent to Fy. This suggests that there is an embedding of Z into X, and this is|eorposition. Let X be he self dual normal N-rigged M-module. Let Ube a
Vomodule W which 1s eerom
is Ther there is a normal

mapping is easily seen to be an M-N-bimodule homomorphism of Z into X which|(namely ’ xO)

N-rigged space Y is a dual Banach space, with the functionals y = p(y", » Jy)|

spanning a norm-dense submanifold of the pre-dual of Y as y' ranges over Y and p
ranges over the pre-dual M, of M [27, Proposition 3.8]. For X = Hom, wn, Fz()):

LL

in fact the case, namely, the mapping z ~ x, where x, (1) = z ® u for all u € U. Thisorpreserves the N-valued inner products. Now Paschke showed that any self-dual

this is particularly apparent (and follows from [28, Theorem 1.4]). Furthermore, it|

product

the two actions commuting

defined by(x,y.

noaduie withsuch that X is isomorphic to Homey, (U, W), with N-valued inner= x*y.

osCo)
representation immediately yields such results as Paschke’s proposition on

polar

decompositions [27, 3.11] (by applying Lemma 1.2).

is easily seen that the image of Z in X is dense for the weak-% topology. Finally, as

in Paschke’s theorem, there is a natural conjugate isomorphism of the N-dual

7. Morita

Homy (Z, N) of Z onto X given by assigning to any 4 € Homy(Z, N) the element+

xy, in X defined byh.

ysettinequi

Morita.

8|

xp, zou)=h(z)u.

We have thus sketched the proof of all but the last statement of the following result,.
)i

In this section we will study normal «-functors which establish an equivalence
between the categories Normod-V and Normod-M. Such functors will, of course, bethi] (thatpective on morphisms), and so we begin by considering when theby a normal N-rigged M-module is faithful.

or definedpart of which generalizes [27, Theorem 3.21:

ooequivalence and its characterization

Jie Proposition. Let M and N be W algebras and let X be a normal N-rigged

6.10. Proposition. Any normal N-rigged Mmodule Z can be embedded as a weak-x|

dense subspace of a self-dual normal N-rigged M-module X which is conjugate iso-|ps - pe N, 0 ' ¢ the weak closur e os og span of the range of the inner productmorphic to the N-dual of Z. As X we can take Hom (U, Fz (U)), where U is any
generator of Normod-U. Also, X is unique up to isomorphism. The functors F; and!

In particular Fre N; Noa je : ten F x faithful i and only if Ng = N,then Fy(V) = {0}if and only if Ny V = {0}.
ormod-N»|

Fy xX will be naturally equivalent.

The last statement of the above proposition follows from:
6.11. Proposition. Let X be a self-dual normal N-rigged M-module and let Z be a

.

i

|
|

module over this ear

sub-M-N-bimodule of X, so that X can be viewed as a normal N-rigged M-module.
Assume further that Z is dense in X for the weak- topology on X (defined above).
Then the functors Fy and F, are equivalent.

.

Proof. It is easily seen that if Ny V'= {0}, then F(V) = {0}. Furthermore, if Ny#N,
then, since Vy, is weakly closed, it is generated by a central idempotent [32, 1.10.5]
and so has a complementary weakly closed two-sided ideal. If V is any normal
ideal, we can view V as a normal N-module, andSuobos x(V) -! ; oe he x is not faithful if Ny # N.
that FAS Somerse A at f ’ € Normod-WV, that f€ Homp, (V, W) and f # 0, but
|

xX

-

We willshow that it follows that , ati|84

M.A. Rieffel, Morita equivalence|

7.2. Lemma. Let M and N be W*-algebras and let F be a normal s-functor from|

M.A. Rieffel, Morita equivalence85

Ube a generator for Normod-N, let ¢(V) = Homy (U, U), and let X = Hom,(U, F(U)),

Normod-V to Normod-M. Let V,W € Normod-N and let f € Homy (V, W). Let thepolar decomposition of f be f = pif |. Then the polar decomposition of F(f) is

so that F is equivalent to F. x as in Theorem 5.5. Since F is faithful, the weak closureof the range of the inner product on X must be all of N by Proposition 7.1.

Proof. It is easily seen that F(1f 1) > 0 and that its square is equal to |for Normod-M. It follows from Proposition 1.3 that M is faithfully represented on
F(U). Furthermore, since F is faithful and full, it establishes an isomorphism of

F(f)y=Fp) ff.

Now since F establishes an equivalence, it follows that F(U) will be a generator

F(f)I2, sothat | F(f)I=F(If1). Now F(pp™) and F(p* p) are both projections, and so F(p) is;

a partial isometry, or must be § pers y ° the partial isometry in the polar
decomposition of F(f), since F(p) | F(f)|= F(f). Now p™p is the support projec-|tion of | 1. It follows that p* p is the weak limit of projections which are dominatedby positive scalar multiples ofacting of Ht or on ieand Min tum can be viewed as the commutan ofacting on- rom
this 1act we can equip A withan additional piece of struc-ture, namely, an M-valued inner product for the left action of M on Xx Specifically,£1. Since F is normal, it follows that F(p*p) is the

weak limit of certain projections dominated by positive scalar multiples of |¢(N) = Homp(U, U) with Hom, (F(U), F(U)), so that c(V) can be viewed as the0)

for x,y € X we see that xy™

F(f)l.

element ofM. We set

puting these facts together
pri that F((p) must be exactly the partial isometry
inthe polar decomposition o.

|

Retuining to the proof of Proposition 7.1, we let f= p | f | be the polar decompo-||

sition of the f for which Fy(f) = 0. It follows from the above lemma that Fy(p) is|(x, phy =x na:
Then it is easily seen that X becomes a left M-rigged space [30, Definition 2.8].
Furthermore, the M- and N-valued inner products are readily seen to satisfy the rela-tion

the partial isometry in the polar decomposition of 0, so that F' x(®) =0. Let V, bethe range of p” p. Since p” p acts as the identity on V, and Fy(p" p) = 0, it follows(x, Php? =x(, Dy

that Fy(¥;) = {0}. In particular, x ® v = 0 as an element of Fy(¥;) for all x EX,|

vE VV), and so:

0=(x®u xe v)=Ux" xvv")

Functors which establish an equivalence of categories will also be full (that is,||: pongo. eh

x,y<cX

Normod-N to Normod-M. If F is faithful and full, then F is automatically normal.
||

faithful and full. But any %-isomorphism of von Neumann algebras is automaticallynormal [7, Corollary
1, 54].) It follows from Proposition 4.7 that F is normal. OJ
i
b

|

|

equivalent if there is an equivalence of Normod-NV with Normod-M implemented by((which will automatically be normal).

We shall now see how W *-algebras which are Morita equivalent are constructedfrom each other. Let M and N be W*-algebras, and let F be a s-functor from
Normod-V to Normod-M, which establishes an equivalence of these categories. Let

2 which mould have the prop

W=p

- wince M canbe viewedthat p(x, y " mnas the commutant of ¢(V),

that oo a 03 then there ht nonzer 0 element x N omen U, "), which on beviewedHom, (F(V), F(V)) will be a x-isomorphism of von Neumann algebras since F is7.4. Definition. Let M and N be W"-algebras. We shall say that M and N are Morita

let

W will be c(V)-invariant. Then, since c¢(IV) is faithfully represented on U so that U
is a generator for Normod-c(V) by Proposition 1.3, it follows from Proposition 1.1

7.3. Proposition. Let M and N be W *-algebras, and let F be a «functor from

(V, V) topy), =xny*

from NV to M is clearly normal, so that X becomes a normal left M-rigged right&functors

nxn,

N-module for the obvious definition. Finally, we show that the weakly closed twosided ideal spanned by the range of the M-valued inner product is all of M. If thiswere not the case, then the complementary two-sided ideal would be generated bysurjective on morphisms between objects in their range).

*

for all x,y,z € X. Also for fixed x,y € X the map|

forallx,x'€X, v,v € V;. It follows that Ny ¥; = {0}. Since V; # {0} (becausep #0), it follows that No # N. O-

Proof. If ¥ € Normod-/V, then the map defined by F from Hom€ Hom, (F(U), F(U)), and so can be viewed as an

as an elementof

X.

Then (x, x); =xx"

canbe viewed

as an element oHom,(W, W). But p acts as the identity on W, so that we would havepix, x)y, = pxx* £0,
.
.
tion.
Thus
p(F(UY) = {0}. But FY
vor for N
nrcontrary to assumption.
Thus p=. Bu
is a generator for Normodand so M is faithfully represented on F(U) by Proposition 1.3. Thus p = 0 as desired.

This leads us to make the following definition, in analogy with [30, Definition

6.10] and [2, Definition 3.2]:|||

7.5. Definition. Let M and N be W *-algebras. By an M-N-equivalence bimodule wemean an M-N-bimodule x which is equipped with M- and N-valued inner productswith respect to which X is a normal N-rigged M-module and a normal left M-rigged
right N-module such that

86

M.A: Rieffel, Morita equivalence(<x, ypz=x(p,2)y forall x,y,z €X;y:(2) the range of ¢ , ),, spans a weakly dense subset of M, and the range of { , Jy

spans a weakly dense subset of »,:We shall say that the equivalence bimodule X is self-dual if it is self-dual both as:

M.A. Rieffel, Morita equivalence=

condition (2) holds. Then, by Proposition 6.12, X is of the form Hom,(U, W) forU a generator of Normod-N and W = Fy(U). By Proposition 7.6, M coincides withE
ES

an N-rigged space and as an M-rigged space.

|

7.6. Proposition. Let M and N be W*-algebras and let X be an M-N-equivalencebimodule. Then, if we view X as an N-rigged space with corresponding algebra Lp (X)of bounded operators, the homomorphism defining the action of M on X is an isomorphism of M onto Ly (X). In particular, L NX) is a Walgebra. Similarly, viewing
X as a left M-rigged space, the antihomomorphism defining the action of
Non Xis|||[an anti-isomorphism of N onto LyX), so that Ly (X) is a W™-algebra.
Proof. Let M(, be the (unclosed) two-sided ideal in M spanned by the range of the
M-valued inner product. By assumption,
My is weakly dense in M. Suppose there isan element m of M such that mX = 0. Then|m x,y dy = (mx, y 0, =0
for all x, y € X. Since Mis weakly dense, it follows that m = 0, so that the homo-87

Lp (X). Note that every element of Hom,(W, W) defines in an obvious way anelement of L p(X), so that we obtain a homomorphism of Hom,py (W, W) into
Ly (X). Now for x,y,z€ X we havex yiyz=xp,2ly=xy°z,
from which it follows that*yh =xy",which is also an element of Hom,(W, W). Since these elements are by assumption|

weakly dense in J, it follows thatM can be viewed as a weakly closed two-sided:

ideal of Hom, (W, W). But X is self-dual as a left-rigged space over Hom(W, W)
by the left-sided version of Theorem 6.5. It follows easily that X is self-dual over M,
so that condition (3), and hence condition (1), is satisfied. A similar argument applies if we start with condition (3). CI|

Let X be an M-N-equivalence bimodule. Then exactly as in [30, Definition 6.17]|

we can form X, the space X but with conjugate operations of M, V and the complex

morphism of M into Ly (X) is injective. Now by the Kaplansky density theorem[7, p. 43] there is a net {my} of elements of M of norm one which converges weakly
to the identity element of M. Let T € Ly (X). Then||

numbers (that is, n¥ = (xn™*)™, (%, Pn =x, yy.etc.). We remark that if X is selfdual, and so of the form Hom,(U, W) as in Proposition 6.11, then there is a natural
identification of X with Hom, (W, U). At any rate, it is easily verified that:

T<x,y wz =Tx (yz Iv ={Txy EEThus My, is in fact an ideal in Ly (X). In particular, Tmy €M,ibimodule.

7.8. Proposition. If X is an M-N-equivalence bimodule, then X isan N-M-equivalence
for each k. Since

I Tm I < I TH, and since any closed ball in M is compact for the weak topology(this being the weak-* topology from the pre-dual), the net { Tm; } will have a subnet|{Tm} weakly convergent to an element my of M. Then for x,y € X we have|theorem for the algebraic case [26, 2, 6].

|

7.9. Theorem. Let M and N be W*-algebras. Suppose that there exists an M-N-equivalence bimodule X. Then M and N are Morita equivalent, with an equivalence between Normod-N and Normod-M being implemented by the functors Fy and Fy.
Conversely, if M and N are Morita equivalent, then there exists an M-N-equivalence

1:

1s

*

—

*

We now come to the main theorem of this paper, which is the analogue of Morita’s_

myx, y y= tim Tm;x, y Jy = lim (myx, Tyly=Ty w= {Txy wowhere the limits are in the weak topology on N. Thus myx = Tx for x €X, so thatthe homomorphism of M into L,/(X) is surjective, and so bijective. A similar argument applies to the antihomomorphism of N into Ly(X). Oi|!

The above result should be compared with [2, Theorem 3.4, part 5).

bimodule X which can be chosen to be self-dual, and which is such that F x and Fy

are naturally equivalent to the functors establishing the Morita equivalence. We ob-

7.7. Proposition. Let X be an M-N-equivalence bimodule. Then the following condi-|

tions are equivalent:

tain in this way a bijection between equivalence classes of equivalences from
Normod-V fo Normod-M and isomorphism classes of self-dual M-N-equivalenceibimodules.

(1) X is self-dual.
(2) X is self-dual as an N-rigged space.

(3) X is self-dual as a left M-rigged space.

|

Proof. The second part of the proof follows from the discussion following Definition

7.4 which motivated our definition of an M-N-equivalence bimodule in Definition 7.5.
The only detail which is not clear from that discussion is that F % is naturally* Proof. By definition, condition (1) implies conditions (2) and (3). Suppose now thatequivalent to the functor from Normod-M to Normod-N establishing the Morita|||

equivalence. But this will follow from the first part of the theorem and the easily

verified fact that the “inverses” of two naturally equivalent equivalence are equivalent.

88

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

The proof of the first part of the theorem is very similar to the proof of {30,89

8.1. Proposition. Let M and N be W™-algebras which are Morita equivalent. Then the

Theorem 6.23]. We must show that Fy © Fg and Fg © Fy are each naturally equivalent to the appropriate identity functor. Now [30, Proposition 6.21] is immediately
applicable. (The fact that we are no longer assuming that the range of the N-valued

centers of M and N are isomorphic.

inner product is norm dense makes no difference.) Thus the functor Fg © Fy isnaturally equivalent to the functor Fy, where Y is the pre-Hermitian NV-rigged
N-module X ®;, X. But in analogy with [30, Lemma 6.22] the map¥ oy = (x, y)y|

Proof. This follows immediately from Proposition 2.1. O|

8.2. Corollary. Two commutative W*-algebras are Morita equivalent if and only ifthey are isomorphic.

is a pre-equivalence [30, Definition 5.6] of Y with the norm closed span NV of therange of the N-valued inner product on X, where NV is viewed as a Hermitian N-rigged|

N-module by virtue of the fact that it is a two-sided ideal in NV. Then the functor Fy
is easily seen to be equivalent to the functor F Ny (as in [30, Lemma 5.7]). But NV,
is weak-* dense in V by assumption, and so F, N, 18 equivalent to Fy; by Proposition
6.11. But we have seen that Fy is equivalent to the identity functor. Thus F5 © F Y|

The next result is an analogue of part of [2, Theorem 3.5(5)].

|. 8.3. Corollary. Let M and N be W*-algebras which are Morita equivalent. Then there
is an isomorphism between their lattice of weakly closed two-sided ideals.

is naturally equivalent to the identity functor. A similar argument works for Fy © Fi.

|

Proof. The weakly closed two-sided ideals of M correspond to the projections in the

The uniqueness up to isomorphism of the self-dual M-N-equivalence bimodule|

center of M, and similarly for N. 0

corresponding to an equivalence class of equivalences follows from the corresponding
part of Theorem 6.9. O|

We now describe how this isomorphism is implemented by a self-dual M-N-equivalence bimodule, in analogy with [2, Theorem 3.5(5)}.

7.10. Corollary. Let N be a W*-algebra and let X be a self-dual N-rigged space therange of whose inner product spans a weakly dense ideal in N. Then N and L(X)
are Morita equivalent. Conversely, if M and N are W*-algebras which are Morita

equivalent, then there is a self-dual N-rigged space X the range of whose inner product|spans a weakly dense ideal in N such that M is isomorphic to L(X).
|
Proof. On X, viewed as a left L(X)-module, define an L(X )-valued inner product by(x, SW z=x(, 2|for x,y,z € X. Then, as in [30, Proposition 6.2, 6.3] it is easily verified that X be-

sided ideals of N (and M) and the lattice of weak-x closed M-N-submodules of X. If
J is such an ideal of N, generated by the central idempotent e, then this isomorphism
is given in one direction by assigning the subspace Xj = Xe to J. In the other direction this isomorphism is given by assigning to a weak-x closed M-N-submodule Y theweakly closed two-sided ideal Iy generated by (Y, Y)y.
Proof. We must show that X Iv = Yand/y =comes an L{X)-N-equivalence bimodule. The converse follows immediately from
Theorem 7.9 and Proposition 7.6. O|

7.11. Corollary. Let K be a Hilbert space, and let L(K) denote the von Neumannialgebra of all bounded operators on K. Then L(K) is Morita equivalent to the one-dimensional W*-algebra, C.

||

Proof. View KX as a self-dual C-rigged space. [])]}8. Properties of Morita equivalence

8.4. Proposition. LetM and N be W*-algebras, and let X be a self-dual M-N-equivalence bimodule. Then there is an isomorphism of the lattice of weakly closed two-J. To show the first equality, let e be

the central projection generating I. Then it is easily seen that y = ye fory €7, sothat Y < Xe. The opposite inclusion is shown by making precise the symbolic calculation
Xe=X(Y, Yon =X, You Y cr.
Conversely, given J, generated by e, we have||

(Xj Xply =(Xe, Xedy =e(X, X)ye ST,
so that /y : CJ. But since (X, X)y is weakly dense in NV by assumption, the opposite
inclusion is shown by making precise the symbolic calculation
J=eNe=e(X, X)ye=(Xe, Xe), c Iy, 0|

imprimitivity bimodule [30, Definition 6.10]. This fact is closely related to Mackey’s
normal subgroup analysis [24], as will be shown elsewhere.

In this section we gather together various general facts about operator algebras

We remark that a similar result is true for C*-algebras which are related by anwhich are Morita equivalent.

90

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

We now investigate how Morita equivalence is related to forming tensor productsof W™.algebras (over the complex numbers, as on [7, p. 24] or [32, Definition
1.22.10] — for an interesting different kind of tensor product see [19D).

91

factor is isomorphic to the algebra of 2 X 2 matrices over itself (which is an algebra
to which it is clearly Morita equivalent). Notice also that Proposition 2.5 shows thata type I factor cannot be Morita equivalent to a type II or type III factor, nor can atype II factor be equivalent to a type III factor.
8.5. Proposition. Let M, M 1 NV, Ny be W*-algebras. Let X be a normal N-rigged|

M-module and Y a normal N-rigged M;-module. Then the algebraic tensor productX ® Y over the complex numbers, with N ® N,-valued inner product defined by.

(xe y, x 8 Ving,

= (x

,

ye

|

,

(yy

and completed for the usual norm, is a normal N @ N-rigged|
M ® M -bimodule. If|

Xand Y are in fact equivalence bimodules and if an Me My-valued inner product’|is defined by
.

,

,

i

(xoy,x'®y") MoM, =(x, x"), (py)
M
M,
then

X © Y (completed) becomes an equivalence bimodule.

and Ny,w

are Morita equivalent.

Proof. The representations of NV on ¥ and W are quasi-equivalent [8, 5.3] and so 8,

31]

ere ae Fibers spaces H and K such that Ve a and re K an

as normal/V-modules. But
the commutant of NV acting on
Ve
His Ny ®
,p. 24], which, by Corollary 8.5, is Morita equivalent to N y- Similarly, N we LK) is.
),
Moritaequivalent
to Vy,. But Ny,, ® L(H) is2 isomorphic) to Ny,, ® L(K). O

8.9. Proposition. If M and N are W*-algebras which are Morita equivalent, then theiropposite algebras are Morita equivalent.
.

Proof. We have not found any way to prove that the indicated inner products are
positive except to imitate the proof for tensor products of ordinary Hilbert spaces,

which involves expressing vectors as linear combinations of orthonormal vectors.
In our setting this means that we must embed X and Y in self-dual modules (Proposition 6.10) so that by Proposition 6.12 we can apply Lemma 1.2 and Lemma 6.7
to express elements of X and Yin terms of orthonormal families of elements. The||

proof of positivity is then carried out pretty much as for ordinary Hilbert spaces.
The rest of the proof of this proposition is carried out by routine calculations. OJ|

8.6. Corollary. Let N be a W*-algebra, let K be a Hilbert space, and let L(K) be the|

von Neumann algebra of all bounded operators on K. Then N © L(K) is Morita
equivalent to N.

|
Proof. View K as an L(K)-C-equivalence bimodule as in Corollary 7.11, and NV as an
N-N-equivalence bimodule.
:
8.7. Corollary. Every type IL, factor is Morita equivalent to a type Il factor. Everytype 1 factor is Morita equivalent to a type I, factor.
Proof. If Vis a type II; factor and if K is an infinite-dimensional Hilbert space, then
Ne L(K) is a type II factor {32, Theorem 2.6.6] to whichN is Morita equivalent
by Corollary 8.6. Conversely, if M is a type II factor, then, as in [7, Exercise 5,
p. 242], M is isomorphic to N & L(K) for some type II; factor N and Hilbert space
KO|

Whether the type II; factor to which a type II_ factor is Morita equivalent is|

unique up to isomorphism is not clear, since it is still unknown whether a type II;8.8. Corollary - Let N be a Walgebra, and let Vand W be two generators for Normod-V.
Let Ny and Ny, denote the commutants of the actions of N on V and W. Then N Vv

Proof. Let M and N denote the opposite algebras of M and NV and let x be an M-Nequivalence bimodule. Then X can be viewed as an N-M-equivalence bimodule bysetting ix =xn, (x, y)gz = ((x, y),)~, etc. O
Wh
W* aleobra i alMor+aleebra isnotether aW"-algebra is always
Morita equiv ent to its opposite algebra is noclear. This is related to the unsettled question of whether a W *-algebra is always
isomorphic to its opposite algebra.

.
8.10. Theorem. 4 W “algebra is of type 1 if and only if it is Morita equivalent to acommutative W*-algebra. In fact, any type W*-algebra is Morita equivalent to its

center.

.
.
Proof. If a W"-algebraM is Morita equivalent to a commutative W *-algebra NV, thenNormod-M is equivalent to Normod-A, and so by Proposition 2.5, M must be oftype I. Suppose conversely that M is a type I W™*-algebra. Then by [7, Theorem 1,
p. 123] there is an Abelian projection e inM whose central support is the identity
element of M. Then eMe is a commutative W *-algebra which by [7, corollary to
Proposition 2, p. 18] coincides with eZe, where Z is the center of M. Now if p is any
projection in Z, then epe 3 0 since e has the identity of M as central support. It
follows that the surjective homomorphism z + eze of Z onto eZe is injective and sois an isomorphism of W™-algebras. Now MeM (the weak closure thereof) is a twosided ideal in M, so generated by some central projection which must dominate e,
and so which must be the identity of M. Thus MeM = M. Finally, let X = Me, anddefine M and eZe-valued inner products on X by(me,*m ely, =mem,,(me, me),Cx=em me.

92

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

Then it is clear that X is an M-eZe-equivalence bimodule, so that M is Morita equiv-

and L(A) is anti-isomorphic to R(4). Thus L(4) is isomorphic, hence Morita equiv-

alent to eZe, and hence to Z. OJ

8.11. Corollary. Two type I W*-algebras are Morita equivalent if and only if theircenters are isomorphic.
8.12. Corollary. Any two type 1 factors are Morita equivalent.

!
|

8.13. Corollary. Any type I W*-algebra is Morita equivalent to its opposite algebra.

|

to M. But ¢(NV) = R(A) so that VN is anti-isomorphic to c¢(V). In other words, NV isisomorphic to the opposite of the commutant of M acting on V. Passing to the real|of the commutant of NV. OJ

dual of V we obtain a normal N-module such that M is isomorphic to the opposite

Let M and N be two type I W*-algebras which are Morita equivalent, so that theircenters are isomorphic. Then we can realize both of their centers as L=(S, u) for an
appropriate measure space [7,32], and, in the separable case, we can form direct
integral decompositions of M and NV over this measure space [7, 32]. Thus we see thatthe way in which M and WV differ is exactly in that their “full matrix algebras”, thatis, type I factors, over each point of S can have different dimensions. Conversely, it
is clear that two W*-algebras constructed by taking direct integrals of type I factorsof possibly differing dimensions but over the same measure space will have isomor-

8.16. Corollary. Let M and N be type 111 von Neumann algebras on separable
Hilbert spaces. If M and N are Morita equivalent, then they are isomorphic.
||

phic centers and so will be Morita equivalent. Thus at least in the separable type I

Proof. Let V'be a generator for Normod-V such that M is isomorphic to the oppositeof the commutant of N on V. Let U be the Hilbert space of a modular Hilbert algebraof which A is.isomorphic to the left algebra. We view U as a normal N-module, whose

commutant is isomorphic to the opposite of N. Since bothM and the opposite of N

case we obtain a good picture of how W *-algebras which are Morita equivalent are
related.

The above considerations are all quite elementary. But by invoking Tomita theory|

we can obtain a quite precise description of Morita equivalence for general W*-algebras
in terms of traditional concepts.
.
8.15. Theorem.* Let M and N be W *-algebras. Then M and N are Morita equivalent||

if and only if there is a generator V for Normod-N such that M is isomorphic to theare countably decomposable and of type II, it follows from [7, Corollary 8, p. 301]
that U and V are isomorphic as N-modules. Then their commutants, which are theopposites of NV and M, are isomorphic, so NV and M are isomorphic. 0.
The above results indicate that for W™-algebras, Morita equivalence is not a fundamentally new concept. For type I algebras it does provide a pleasant point of view.
In fact it suggests that perhaps a neater way to define type I algebras is simply to

say that they are the ones which are Morita equivalent to commutative ones. On the

opposite algebra of the commutant of N acting on V.

|

Proof. Let V be a generator for Normod-¥, and let N' denote the commutant of N|

acting on V. By [35, Theorem 12.2] N is isomorphic to the left algebra L(4) ofsome modular Hilbert algebra A. By the commutation theorem for left Hilbert al-other hand, in the type II case the fact that Morita equivalence seems to be weaker
than isomorphism may possibly prove useful since isomorphism seems so intractable there — witness the questions of whether a type II; factor is isomorphic to

2X 2 matrices over itself. It might, for example, be possible to obtain at least a
partial classification of Morita equivalence classes of type II factors. A number ofgebras [35, Theorem 4.1], the commutant of L(4) is R(4), the right algebra of 4,

the invariants which have already been introduced for factors will probably turn

* Alain Connes and Masamichi Takesaki have pointed out that for factors on separable Hilbertiivalencet
ut to be the same as the notion of genus introduced b|

Neumann (26a, Section 3]. Alain Connes has also bointod out that ed

W -algebras are Morita equivalent if and only if they have faithful representations with isomorphic commutants.

alent, to the opposite algebra of R(4). Since the representations of N on both V andthe Hilbert space U of 4 are faithful, and thus are generators for Normod-N, we canapply Corollary 8.8 and Proposition 8.9 to conclude that NV is Morita equivalent to
the opposite algebra of N'.
Suppose now that M is isomorphic to the opposite algebra of N'. It follows that
M and N are Morita equivalent. Conversely suppose thatM and V are Morita equivalent. Choose U as above, so that R(4) is anti-isomorphic to V. Then the argumentsin the second paragraph after Definition 7.4 show that, if we adopt the notation

used there, F(U) is a faithful normal c(V)-module whose commutant is isomorphic

8.14. Corollary. Any type 1 von Neumann algebra is Morita equivalent to its commutant.

Myrray ve

93

out to be, in fact, invariants for Morita equivalence.

classes of factors.
.

8.17. Definition. Let 4 and B be C™algebras. We say that 4 and B are Mor ia||

equivalent if there is an equivalence of Hermod-B with Hermod-A which is implemented by #-functors (which will automatically be normal).

|

8.18. Proposition. The C*-algebras A and B are Morita equivalent if and only iftheir enveloping W*-algebras n(A) and n(B) are Morita equivalent.

94

M.A. Rieffel, Morita equivalence

M.A. Rieffel, Morita equivalence

It follows that the earlier results of this section are immediately applicable to{31 po Alacra

C*-algebras, although they do not, in general, have nice formulations in terms of the
:

Toros

wn

95

on lo the nor separable case of Takesaki’s duality theorem forC*-algebras themselves rather than the corresponding W enveloping algebras.

[4] N. Bourbaki, Topologie Generale, Actualités Sci. Indust. 1045, (Hermann, Paris, 1958)

The detailed study of Morita equivalence for special classes of C" -algebras must
R.
.
:
await another time. e. We include here only the following result.

[5] C. Chevalley, Theory of Lie Groups (Princeton Univ. Press, Princeton, N.J., 1946).
.M. Col Cohn, Morita equivalence[6] P.Mq: and dualit Ys Queen Mary Collegeg Math. Notes, , London,

y

8

Co

:

.

ch. 9.

.

.

[7] J. Dixmier, Les Algebres d’Operateurs dans I'Espace Hilbertien, 2d ed. (Gauthier-Villars,.

Paris, 1969).

19. Proposition. . Two commutative separable
8.19.
C*-algebras are Morita equivalent if-(8] J. . Dixmier, , Les Caleb.
.
:
gtbres et leurs . Representations,2d ed. (Gauthier-Villars,
Paris, 1969).

and only if their spectra are Borel isomorphic.
!

Proof. Any two commutative C*-algebras (separable or not) which have Borel iso;iimorphic spectra will; be Morita
equivalent, as can bes on from|

Example 3.11. Con-

arch eres

metric’force sol onais sp > es [ 4

p

122]

br

i

ing

i

i

preserving isomorphism

of[17] P. Freyd, Abelian Categories (Harper and Row, New Yo.k, 1964).

&[18] L.T. Gardner, On the “third definition” of the topology on the spectrum of a C*algebra,
Can. J. Math, 23 (1971) 445450.

[19] A. Guichardet, Sur la catégorie des algebres de von Neumann, Bull. Sci. Math. 90 (1966)
41-64.

[20].E. Hewitt and K.A. Ross, The Tannaka—Krein duality theorems, Jber. Deutsch. Math.-Verein.

onto M(Y'). One can conclude from this also that

MX)

But

( )-O t clear how

much

more

TI%0

(Thompson, Washington, D.C., 1967) 192-2135.
[15] J. Ernest, The enveloping algebra of a covariant system, Commun. Math. Phys., to appear.
[16] J. Ernest, A duality theorem for the automorphism group of a covariant system, Commun.
Math. Phys., to appear.

mutative C*-algebras 4 and B are Morita equivalent,be isomorphic. (The pre-dual of any W-algebra is unique — see [32, p. 29].) Thus,ar elds statistics and non-Abelian gauge groups, Commun.

[14] J. Ernest, Hopf—von-Neumann algebras, in: Proc. Functional Analysis Conf., Irvine'Bt al
so that n(4) is isomorphic to n(B), then the pre-duals of n(4) and n(B)must also

if X and Y are the spectra of A and B, and M(X) and M(Y) denote the spaces ofregular Borel measures on X and Y respectively, there will be an isometric order-Fhys.

[12] J. Ernest, A new group algebra for locally compact groups, Am. J. Math. 86 (1964) 467-492.
[13] J. Ernest, A new group algebra for locally compact groups II, Can. J. Math. 17 (1965)
604-615,

are those of standard Borel Sp aces [1 ’ Proposition 2.3]. But any two standard Borel
spaces of the same cardinality are isomorphic [1, Proposition 2.7]. (J.
We remark that if two com,

Math. Phys. 23 (1971) 199-230.

[10] y th Pre 28 (Lorn[11] S. Eilenberg, Abstract description of some basic functors, J. IndianMath. Soc. 24 (1960)
231-234.

and so their Borel structures’[91 S. Doplicher, R. Haag and J.E. Roberts, Local observables and particle statistics I, Commun,ath.

hi o are Morita

versely, let A and B be two separable commutative-algebras whic
equivalent. Then there will be a bijection between their equivalence classes of irreducible representations, and so between their spectra. Thus their spectra have theinality. Sid B are separable, their spectra are separable locally compactn.d.

71 (1969) 61-83.

one can con-

[21]

I Kaplansky, Modules over operator algebras, Trans. Am. Math. Soc. 75 (1953)
839-858.

X and Y have the same cardinality. u 1
163.13 shthat the bijection
clude in general about X and Y. Certainly Example 3.1
showsj[22] G.I Kac, Ring-groups and the principle of duality I, Tr. Moskov. Mat. Obs. 12 (1963)
259-300.

of X with Y corresponding to a Morita equivalence can be very badly behaved.

[23] G.W. Mackey, Induced representations of locally compact groups I, Ann. Math. 55 (1952)

The above considerations indicate that for C*-algebras the notion of Morita
equivalence as defined in Definition 8.16 is probably too weak to be very interesting.

Fa

i

i

r C-algebras which

Rather, it will be strengthened forms of Morita : quivtlence forwill provide the more interesting and useful tools —101-139.

|

M CE

for example,olequivalencesq[25] S. MacLane, Categories for the Working Mathematician (Springer, Berlin, 1971).

oo[26] K. Morita, Duality for modules and its applications to the theory of rings with minimum

which preserve direct integrals, or weak containment, as do those defined by an im-

condition, Tokyo Kydiku Daigaku (A) 6 (1958) 83-142.

primitivity bimodule [30, Proposition 6.26]. Such strengthened forms of Morita
equivalence will play an important role in, among other things, the classification of
C *.algebras, but discussion of this matter must await another tim».

.

[262] Murray and von Neumann, Rings of operators VI, Ann. Math. 44 (1943) 716-808.
[27] le Paschice Inner product modules over B*-algebras, Trans. Am. Math. Soc. 182 (1973)

e.

+

:

[24] sw. Mackey, Unitary representations of group extensions I, Acta Math. 99 (1958)—.

[28] M.A. Rieffel, Multipliers and tensor products of LP-spaces of locally compact groups,
Studia Math. 33 (1969) 71-82.

|’

References:

5(1972)[1] L. Auslander and C.C. Moore, Unitary representations of solvable Lie groups, Mem. . Am.
Am
Math. Soc. 62 (1966).
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R:[29] M.A. Rieffel, Unitary representations induced from compact subgroups, Studia
Math. 42
145-175.

[30] M.A. Rieffel, Induced representations of C *-algebras, Advan. Math. 13 (1974) 176-257.

[ S. Sakai, C*-algebras and W*-algebras (Springer,(1972) 417-430.
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Berlin, 1971).
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N. Saavedra Rivano, Categories tannakiennes, Bull. Soc. Math. France 100

370-382.

.

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M. Takesaki, A characterization of group algebras as a converse of Tannaka—Stinespring—
Tatsuuma duality theorem, Am. J. Math. 91 (1969) 529-564.
[35] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture
Notes in Math. 128 (Springer, Berlin, 1970).
[36] T. Tannaka, Uber den dualitit der nicht-kommutativen topologischen Gruppen, Tohokuf1:Math. J. 45 (1938) 1-12.

[37]

]

N. Tatsuuma, A duality theorem for locally compact groups, J. Math. Kyoto Univ. 6 (1967)

i

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