Fields Institute Communications
Volume 13, 1997

On Selfdual Hilbert Modules
David P. Blecher
Department of Mathematics
University of Houston
Houston, TX 77204-3476 USA

Abstract. We establish some connections between operator space theory, and the theory of selfdual modules.
Every selfdual C*—module
over a W*—algebra is a dual operator space in a canonical way. We findappropriate W*—versions of some results from our previous paper “A
new approach to Hilbert C* —modules”. For instance we prove that thecomposition Y ®¢Z of correspondences Y and Z, is completely isometrically isomorphic, via a W*-homeomorphism, to the “extended Haagerup
tensor product” of ¥ and Z. Also we can identify the (unique) predual of this composition as the module operator space projective tensor
product of Z., the operator space predual of Z, and Y, the conjugate
module of Y. This predual is also completely isometrically isomorphic
to the module operator space projective tensor product of N,, Z andY, where N, is the predual of the W*—algebra with respect to which Z
is a C*-module. These are not true with the Banach module projective
tensor product.

1.

Introduction

Selfdual Hilbert C* —modules over a W*—algebra, which we shall abbreviate inthis paper to W*—modules, were defined and developed by W. L. Paschke [1973,1976], with later contributions by M. A. Rieffel {1974a, 1982]. More recently, they
have played a significant role in noncommutative topology and geometry, under
different guises (for instance in the study of correspondences, or the basic construc-tion of Jones).

A (right) W*—module Y over a W*—algebra M then, is a rightC*—module over M which is selfdual. The last term means that every boundedright M-module map f : Y — M is of the form f(y) =< z|y >, for some z € Y.
Selfdual modules behave almost exactly like Hilbert spaces (as opposed to general C*—modules); for instance there exist orthonormal bases, Gram-Schmidt, theParseval identity, and a good orthogonality theory. There are several nice characterizations of W*—modules in the literature. In this paper we show how the ideas
1991 Mathematics Subject Classification. Primary 46L10,46 M05; Secondary 47D15,46A32.
Supported by a grant from the National Science Foundation.
©

65

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of Blecher [1995] translate into the W*—module setting.

David

P. Blecher

Since W*—modules are

so well behaved, and in particular have orthonormal bases, our methods do notyield as many new results as in the C*—module case. Indeed, after submitting andcirculating an earlier version of this paper we received a reprint of a recent paperof Denizeau and Havet [1994], which contained several of our corollaries. Since ourmethods are different and give another approach to some of their results, we havefor the most part not removed these overlaps, but indicate where they occur.
We shall assume that the reader is familiar with the basic ideas explained in
Blecher [1995]. In particular, Section 2 there summarizes most of the necessary
background and notation on operator spaces and operator modules which we shallneed. Every W*—module is a dual Banach space (Paschke [1973]), we will obtain
versions of the results in Blecher [1995] which are compatible with the weak*topology. The main difference is that the “Haagerup tensor product” is replacedwith the “extended Haagerup tensor product” which was developed in Magajna[1994a] (Magajna originally called this the full Haagerup tensor product, but hehas informed us that he has changed the name). This is the module version of
the “weak*-Haagerup tensor product” of Blecher and Smith [1992]. The definition
and relevant properties of the ‘extended’ Haagerup tensor product are summarised
below in Section 2, along with other preliminaries. We also list there the connections
between operator spaces and W*—modules (and their preduals).
In Section 3 below we describe in operator space language the ‘composition’ of
correspondences. For instance we prove that the composition Y®gZ of correspondences Y and Z, is completely isometrically isomorphic, via a w*-homeomorphism,
to the extended Haagerup tensor product of Y and Z. We give another proof of
Denizeau and Havet’s result that this composition is isometrically isomorphic, via
the natural map, to the space CBy(Y, Z) of completely bounded module maps
from Y to Z. In fact this is a completely isometric isomorphism. From this we
get a useful explicit form of the predual operator space of Y®¢Z. Namely, it iscompletely isometrically isomorphic to the module operator space projective tensor product of Z,, the operator space predual of Z, and Y, the conjugate moduleof Y. Thus it is also completely isometrically isomorphic to the module operator
space projective tensor product of N,, Z and Y, where N, is the predual of theW*—algebra w.r.t. which Z is a C*-module. These are not true with the Banach
module projective tensor product. We also define a W*—module version of theexterior tensor product of C*—modules, and show that this may be identified completely isometrically with CB(Y,, Z), for W*—modules Y and Z. In Section 3 wealso prove the basic HOM-tensor relations for W*—modules, and get better results
than in Blecher [1995]. One example of these relations, (although this one is moreor less in Blecher [1995]), is that the bounded module maps By (Y®yZ, W) froma composition of correspondences Y and Z into a third W*—module W, is completely isometrically isomorphic to Bas (Y, By (Z, W)) (where Bs are the completely
bounded module maps - see Section 2). In Section 4 we make some remarks on
the Eilenberg-Watts and fundamental Morita theorem for W* —module categories.
Many of the ideas contained herein may be found in Paschke [1973, 1976], Rieffel(1974a], Ghez, Lima and Roberts [1985], and Denizeau and Havet [1994], and someof what we do below can be construed as rephrasing part of what these authors
did in operator space language. However, it is to be hoped that the operator spacemachinery will prove rewarding, and we do obtain many new results.

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We thank Jon Kraus for several conversations, and the referee for useful comments.

2

Preliminaries

A dual operator space is an operator space which is completely isometric to thestandard dual (see Blecher [1992a], Blecher and Paulsen [1991] or Effros and Ruan
(1991a]) of some other operator space. In Blecher [1992a] the reader may find a simple proof of the fact that a dual operator space is w*-homeomorphic and completely

isometric to a weak*-closed subspace of B(H) (the predual of this embedding is explicitly listed after Proposition 3.1 there).

B(H)Conversely, a weak*-closed subspace ofis a dual operator space (this is proved in Blecher [1992a] for von Neumann

algebras, but by using this in conjunction with Corollary 2.4 there, we get it for
any weak*-closed subspace). A w*-homeomorphism between dual spaces is a homeomorphism for the weak*-topologies. Similarly, a w*-isometry or w*-isomorphism
shall mean an isometry or isomorphism which is a w*-homeomorphism. A normal*_-homomorphism from one W*—algebra to another is assumed to be unital, andwill also be referred to as a normal representation.
If Y is an operator space, and I, J are cardinal numbers, then we write Mj j(Y')
for the set of I x J matrices whose finite submatrices have uniformly bounded norm

(see Effros and Ruan [1988, 1990]). Such a matrix is normed by the supremumof the norms of its finite submatrices. We write M;(Y) = M; 1(Y),CF(Y) =Mra(Y),

and RY(Y)=M, ;(Y).

We shall write (y;)ic; and[y:]ics for typicalelements of C}(Y) and RY (Y) respectively. Generally [-] denotes a row, and (-) acolumn. We write C;(Y") (resp. R;(Y")) for the subspace of Cy (Y) (resp. RY(Y))
which

is the norm

closure of the set of elementswith only a finite numberofnonzero entries. If Y = C then Cy (Y) = C;(Y), and we write this column Hilbert
space as Cr. Similarly, Ry = R;(C). In Wittstock [1980], Blecher, Muhly andPaulsen [1994], Blecher [1995], or Magajna [1994b], and elsewhere, it is explained
how every C*—module has a canonical operator space structure, as may be seen, forexample, by viewing it as the 2-1 corner of the linking C*—algebra. Similarly, every
W*—module Y over a W*—algebra M is the 2-1 corner of the linking W*-—algebraof 2 x 2 matrices
a Zz

y b )
forae M,yeY,z€Y,beB(Y).

as B(M ®Y)This linking W*—algebra may also be regarded(which is defined later). Since a W*—module is thus a corner of aW*—algebra, it follows by duality principles that its predual is unique. If Y, isthe Banach space predual of a W*—module Y then it inherits an operator spacestructure from Y™*, and with this operator space structure, (Y,)* = Y completely
isometrically. Indeed, using canonical operator space identifications (see Blecher

and Paulsen [1991] or Effros and Ruan [1991a]), we can write down the predual
operator space explicitly as
Y.

=

M,

QM

Y,

(H

where M, is the operator space predual of M, Y is the conjugate W*—module of Y,,
and ®)s is the module operator space projective tensor product. For completenesswe give the easy argument. We refer the reader to Blecher and Paulsen [1991] orEffros and Ruan [1991a) for the definition of the operator space projective tensor
product; the latter contains the ‘internal’ norm formula for this space - we shall notDavid68

P. Blecher

need this formula explicitly here, but it is nice to know that such a formula exists.
We recall (see Blecher [1993, inpreparation]) that if V and W are respectively left
and right operator modules over A then V ® 4 W may be defined to be the quotientof the operator space projective tensor product (see Blecher and Paulsen [1991] orEffros and Ruan [1991a]) V ® W by the closed subspace generated by elementsv® aw — va ® w. As in Blecher, Muhly and Paulsen [1994] it can be shown thatthe module operator space projective tensor product is associative, and it linearizes
jointly completely bounded balanced bilinear maps. Thus by definition M, ® mY
is a complete quotient of M, ® Y. Hence by duality, (M, um Y)* c (M. ® Y)*
completely isometrically. However the latter space is CB(Y,M) by Blecher andPaulsen [1991] or Effros and Ruan [1991a], and it is straightforward that the rangeof the inclusion above consists of M-module maps. Conversely, any c.b. M-module
map Y — M obviously defines an element of (M, ® um Y)*. This proves (1), sinceCBu(Y,M) = By (Y,M) 2 Y completely isometrically (see next paragraph andBlecher [1995]). More generally, if Y is a C*—module over a W*—algebra M thenthe operator space predual of the selfdual completion of Vis M, ® mY. We remarkthat the selfdual completion of a C*—module over a W*—algebra M, as defined
to be a W*—module over M containing Y as a weak*-dense submodule, is unique(see Rieffel [1974a], Theorem 6.10). If T': Y — Z is a surjective isometric module

map between W*—modules over M, then T is unitary (see Lance (1995) or Blecher[1995]). Also, T is a w*-homeomorphism, and the unique preduals of Y and Z arecompletely isometrically isomorphic via the module map 7.
If Y is a right W*—module over M, and if Z is a right operator module over M

in the sense explained in Effros and Ruan [1988] or Blecher [1995], then we write
B(Y, Z) or By (Y, Z) for the space CBum(Y, Z) of completely bounded module maps

Y — Z. If Z is also a C*—module over M then all bounded module maps Y — Z
are adjointable, with norm = c¢.b. norm, so that in this case B(Y, Z) coincides withBum (Y, Z) isometrically, and thus it means the traditional thing. As explained in
Blecher [1995] for example, B(Y, Z) is an operator space. Moreover, if Z is alsoa W*—module then B(Y, Z) is a dual operator space; the operator space predual(which again is unique, since B(Y, Z) is a corner of the W*-algebra B(Y & Z)) is

easily shown by an argument similar to that for ({) above, to be

B(Y,Z), 2Y ®u Z. 2Y un Mi. ®u Z

(11)completely isometrically. The following standard result shall be used several times
without comment: a bounded net Ty in B(Y, Z) converges in the weak*—topology
to T € B(Y, Z) iff it converges to T in the point weak* topology. This may be seenusing the first 2 in (11). Indeed this =, and consequently also the net convergence
result is true if Y is any right operator module and if Z is a dual operator module,
by which we mean an operator module which is a dual operator space such that themodule action is separately weak*-continuous in each variable. For such a module,Z, is also a module with the obvious action, and is a submodule of Z*.
The notation B(Y, Z) for the completely bounded module maps is convenient
for this paper, but in the case that Z is not a C*—module it is perhaps not sucha good notation. For instance, the reader should be warned that it does disagreewith the notation used in the last identity of Section 8 of Blecher [1995].

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Since pairs of W*—algebras answer the slice map problem affirmatively (seeKraus [1991]), so do pairs of W*—modules, being corners. From this we conclude

by (Blecher [1992b], Theorem 2.5) (the part of this theorem which we are using
here was also proved independently by Ruan) that for two W*—modules Y and Z
we haveY®Z 2 (Y, ® Z,)*

(tt1)

completely isometrically and w*-homeomorphically, where ® is the operator space
projective tensor product mentioned above, and Y®Z is the weak*-closure of Y ® Z
in the von Neumann algebra spatial tensor product of any containing von Neumann
algebras. This formula shows that as a dual operator space, Y ®Z is independent of
the particular containing von Neumann algebras, and we call it the weak*-spatial
tensor product of Y and Z. We see in Section 3 that Y®Z is also a W*—module.

It is well known (see Effros and Ruan [1988]) that C¥(Y) = C;®Y and RY (Y) &R;®Y as dual operator spaces, if Y is a dual operator space.
If {Y,} is a collection of W*—modules, then we shall write ®3Y,

for their

W*—module direct sum (called the ultraweak direct sum in Paschke [1973]). If Y
is a W*—module then what we called C¥(Y) earlier coincides with the ultraweak
direct sum of I copies of Y, for any cardinal I.
We say that a submodule Z of a C*—module Y is 1-complemented if it is therange of a contractive idempotent module map P. It follows (see Blecher [95] for
instance) that P is an adjointable projection in B(Y"), and that there exists anotherclosed submodule W of Y which is orthogonal to Z, with Y = Z + W, so that Y
is the internal C*—module direct sum Z & W. Thus ‘1-complemented’ is the sameas the usual C*—module notion of a complemented submodule. We express it thisway so as to remove the inner product from view. If Y is a W*—module then a1-complemented submodule is also a W*—module, by an obvious argument. The
following well known characterization of W*—modules corresponds to Theorem 3.1

in Blecher [1995]:
Theorem 2.1 Suppose Y is a Banach space and a right module over a W*—
algebra M. Then the following are equivalent(1) Y is a W*—module,(ii)

There exists an index set I and contractive M-module maps ¢ : Y — CY (M)and ¢ : CY (M) — Y such that Y¢ = Idy,(iii) Y

4s isometrically module

isomorphic

to a I-complemented

submoduleofC¥(M) for some set I.
In this case, the inner product on Y is < y|lz >=< ¢(y)|p(2) >, if ¢ is as in (4).
Proof That (i) implies (ii) and (iii) is proved in Theorem 3.12 of Paschke[1973]. That (iii) implies (i) is a consequence of the fact mentioned above that acomplemented submodule of a W*—module is a W*—module. That (ii) implies (iii)
is clear. O

Paschke proves Theorem 3.12 (Paschke [1973]) by first showing that a W*—
moduleY

hasan o.n.b.,thatis, a subset{zs}acs

of ¥

which

is maximal

w.r.t.

the properties 1) < T,|To > is a projection in M, and 2) < zqlxg >= 0if a # SB.
It then follows that y = }__ Za < Zo|y > in the weak*-topology for all y € Y, once

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one establishes the Bessels and Parseval identities.

P. Blecher

We shall refer to this o.n.b.

{4} in the sequel. The element [Zz] of RY (M) clearly has norm 1. Then we may
take #(y) = (< Zo,y >) and P((Ma)) = >, Tame in (il) above. The reader isreferred to Denizeau and Havet [1994] for the related, but slightly weaker, notion ofa quasi-basis: if ¢,1) are as in (ii) above, and if {es} is the usual basis of C}'(M),then {1(eq)} is a quasi-basis for Y.
Most of the following is in Paschke [1973], before Theorem 3.12 there (see also
Ghez et al. [1985]):

Theorem 2.2 Suppose that {Y,}ner is a collection of W*—modules over M,
that Y is a fited W*—module over M, and that there exist contractive M module
maps in : Yp = Y, mp: Y = Yy, with mp 04m = bpm Idy,, for all n,m. Here bpm 18
the Kronecker delta. Then ik = my, and Y is unitarily equivalent to the W*—module
direct sumZ@ (®¥Y,),

where

Z

is

a W*—module

over M.

IfY, inmn

= Idy

in

the weak*-topology of B(Y), then Z = (0).
Proof The ranges 4,(Y,) are mutually orthogonal 1-complemented submodules of Y. The sum Y_ 4,7, is a increasing net of contractive projections in B(Y),indexed by the finite subsets of I directed upwards by inclusion. Hence it con-verges in the weak*-topology in B(Y) to a contractive projection P in B(Y). Let
Z = Ran(I — P). The obvious map Z ® (®%Y,) — Y is well defined, isometric andsurjective (see Paschke [1973]), hence unitary. O

We shall need the weak*-Haagerup tensor product which we introduced in
Blecher and Smith [1992], and its module version which was developed in Magajna[1994a). We defined the weak*-Haagerup tensor product for dual operator spaces,
but it has been pointed out subsequently by several authors (see particularly Effros

and Ruan [1992], Effros, Kraus and Ruan [1993]) that it makes sense for general

operator spaces. Indeed the module version of this tensor product was defined by
Magajna on the class of “full operator modules”. Unfortunately, in the present
context, Magajna’s nomenclature was bound to be confused with the usual notionof a full C*—module, so that we believe that he has changed the name to “strong
operator module”. We now describe this notion.

A subspace Y of B(H) which is right invariant under multiplication from a

von Neumann algebra M C B(H) is called a ‘strong’ module over M if 3° ynmn
converges strongly (or equivalently, in the weak*-topology) in B(H) to an elementof Y, whenever [yn], € R¥(Y), and (my), € CP (M). More generally, if Y is anoperator module over a W*~algebra M, then we say that Y is a strong operator
module (or simply a strong module) over M if(i) there exists a linear completely isometry ® : Y — B(H), and a normal
representation ¢ : M — B(H) such that ®(y - m) = ®(y)¢(m) for all y €Y,m € M, and(ii) ®(Y) is a strong module in the above sense over ¢(M).
If (i) holds then we say that (®, ¢) is a representation of (Y, M), and Magajna provesthat if (ii) holds for some representation then it holds for all representations, andthat ¢ may be taken to be faithful. The class of strong modules over the scalars C
is exactly the class of (norm closed) operator spaces. For most of the applicationsin this paper, Y will be a dual operator space. In fact ¥ will mostly be a dual
operator module, from which it follows (from the proof of Theorem 3.4 in Effros

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and Ruan [1988]) that there is a representation (®,¢) as above for which ® and¢ are w*-homeomorphisms.
Thus a dual operator module is a strong operator
module. Nonetheless, it will be convenient to summarize some results which were
established by Magajna for general strong operator modules.
Suppose that Y is a strong right module over a W* —algebra M, and that Z isa strong left module over M. Let I be a cardinal number bigger than that of some

Hilbert space on which Y and Z can be represented in the sense of (i) above. One
may define the “extended Haagerup tensor product”(1994a]) Y®ra(see Remark 3.7 in Magajna

Z to be equivalence classes of pairs (y,2) € R¥(Y) x C¥(Z), where

two pairs (y,2) and (z,w) are equivalent iff there exists a projection p €Myr (M)

such that [y,z]p = 0 and p* (2, ~w) = 0. Denote the equivalence class of (y, z) as
yOu z. The sum yO pr z+xO pw is defined to be [y, z]Op(z, w) (by the earlier statedconvention, (y, x] means, for example, the element in RY, (Y) given by y followed byz). Finally, we norm the equivalence class of (y,2) by inf{||z||||w||}, the infimum
taken over all equivalent pairs (z,w). A similar formula defines M,,(Y®npZ),
making Y®pZ into an operator space.
A pair (y, 2) should be regarded as a formal sum }, yx ® zx, if y = [yk]k, 2 =(zx)x- We shall see that in the cases which we are interested in, when Y is a
W*—module and Z a dual operator module, then Y®p ps Z is a dual operator space,and has as predual operator space Z, ® wm Y, and the formal sums above defining
the extended Haagerup tensor product converge in the weak*-topology.
Another equivalent way of defining the extended Haagerup tensor productY ®napZ, which is perhaps more practical, is as follows. We suppose that Y and Z

are represented in the sense of (i) above in the same B(H), which is always possible. Let [yn], € RY(Y) and (2,)n € Cy (Z). Identify the formal sum }, yn ® 2,with the operator in CB(M’, B(H)) given by r' — > y,r'2,. We identify twosuch formal sums if they give the same operator in CB(M’, B(H)), and we normsuch a formal sum by the c.b. norm of the associated operator, which turns out tobe the same as inf{|| }_, Yay 2 | 3, 252,]|2}, the infimum taken over all formalsums which represent the same operator in CB(M’, B(H)). (The reader may be
confused at the mention of ‘*’ and multiplication in an operator space, however
it turns out that these expressions give the same number when performed in any

containing W*—algebras, in fact for instance ||)" 2%2n|2 is simply the norm of(2n)ner in CP(Z).) W.lLo.g. we can take I in these formal sums to be the cardinality of the Hilbert space H. The space of such formal sums, up to identification

as operators in CB(M’, B(H)), is denoted by Y®parZ.

Finally, it turns out thatY®numZ is independent, up to completely isometric isomorphism, of the way that

Y and Z are represented in the same B(H).
Magajna remarks that Y®p ps Z contains the module Haagerup tensor productY ®nm Z completely isometrically. If M = C, and if Y and Z are dual operator
spaces, then Y®;, pr Z is exactly the weak*-Haagerup tensor product of Blecher andSmith [1992]. We remark that these definitions may seem cumbersome at first, but
in practice it allows us to get our hands on elements in a very convenient way. These
expressions may be manipulated using slice maps, or using projection techniquesin the W*—algebra or its amplifications.
Magajna shows that if, further, Z is a strong right module over a W*—algebraN, then Y®pZ is again a strong right operator module. Moreover the tensor

P. Blecher

David

72

product is ASSOCIATIVE:(Y®rMmZ)RrNW 2 Y Rpm (ZQrnW)
completely isometrically. The following result is mentioned in Magajna
and may be easily proved by the methods of that paper.

[1994a),Theorem 2.3 (The strong Haagerup tensor product is FUNCTORIAL). If
T, : X; — Y1 is a completely bounded right M-module map between strong right M modules, if Ty : Xo — Y3 is a completely bounded left M-module map between strong
left M-modules,

then Ty ® Ty

is completely

bounded from Xi®um Xo—Yi®pumYa,

and moreover ||T1 ® Taleo < ||T1 leo T2]|coHere T) ® T; is the map Y_; zi ® 2x — yp T1 (zk) ® To(2x), where }, Tx ® 2k
is one of the typical sums

in X;®pp

Xo(that this is well defined emerges from theproof).
The following result, which is essentially Theorem 4.2 and the remark afterwards in Effros and Ruan [1988], is used to prove the above, and also used later inour paper:
Theorem

2.4 If T : Y—

Z is a completely bounded module

map

betweenstrong operator modules, then 3" T(yn)mn = T(3X, ynmx), for all I and [y,] €R¥(Y), and (my) € CY (M).

If Y is an operator space then C¥(Y) = C;®,Y and RY(Y) = Y®uRy,
and C;@,Y®rR; = M;(Y) completely isometrically, via the canonical identifications. Thus, for instance, if Z is a strong left M-module then C7 (M)®nrmZ =C¥(Z) completely isometrically. This is because CY (M)®nmZ = Cr@nMOrmZ =Ci®nZ = CF (2).
This completes our summary of pertinent facts about the strong Haagerup tensor product. For further details see Magajna [1994a]. We now return our attention
to W*—modules.
Every W*—module is a strong operator module in the above
sense, as may be seen by viewing it as a subspace of the linking W*—algebra. Similarly every M — N correspondence (see Section 3) is strong as a left and as a right
operator module, and in fact is a dual left and a dual right operator module. Thus
the important modules in the theory of selfdual Hilbert modules are strong in theabove sense.
If Y is a W*—module over M, and if Z is a strong right module over M, thenthere is a natural inclusion of Z ®
Y into B(Y, Z). The following corresponds to

Theorem 3.10 in Blecher [1995]:
Theorem 2.5 If Y is a right W*—module over M, and if Z is a strong left
(resp. right) module over M, thenY @nmZ = CBy (Y, Z) completely isometrically(resp. Z ®pmY = CBuM(Y, Z) completely isometrically), via the canonical map. If
in addition Z is a W*—module with respect to the module action above, then theseidentifications are also w*-homeomorphisms. Putting Z =Y, we have that Zn
Zwith the natural multiplication(z09)(# ®y) =z <yld >is a W*—algebra (2 B(Z)).

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Proof Define a map © : Z ®puY
— CBy(Y,Z), by O(u)(y) =donzn < Yn, y >, if u = 3 zn ® yo. Note that O(u)(y) = mo (Idz ® gy)(u),where m is the module action Z&,yM — Z, and g, : Y — M : T —< zly >.
This shows that © is well defined. It is easy to see that © is contractive, andindeed completely contractive. Now if T' € CB
(Y, Z), define u = 5° Tea) ® €q,
where {ey} is an on.b. for Y. Since T is completely bounded it follows from
Theorem 2.3 (the functoriality of the tensor product), that u is a well-defined el-ement of Z @amY. Clearly, [ull < | Tllesll 3, ea ® €allyg,,, v < IITllcs- For thisu we have that ©(u)(y) = 3 , T(x) < ealy >= TX ea < ely >) = T(y),using Theorem 2.4. This shows that © is onto, and an isometry. Showing that© is a complete isometry is similar, or it may be deduced from the isometry bya standard argument. If now Y,Z are right W*—modules, then to prove the w*-

homeomorphism Z ®p,aY = CB

(Y, Z) we may assume by taking the direct sum

if necessary, that Y = Z. The space Z®pprZ has (unique) predual by the above.
It is easily checked that with the natural multiplication described in the statement,
the identification Z®nyZ = B(Z) described above, is a homomorphism. Thus,
Z®nmZ is a W*—algebra with this multiplication. By the uniqueness of predual
this identification is necessarily a w*-homeomorphism.
{J
As consequences of the previous theorem we obtain the following useful facts:(1) if Z is also a strong right (resp. left) module over N, then CBy(Y,Z) =Bu(Y,Z) (resp. CBm(Y,Z) = Bp(Y,Z)) is a strong right (resp. left)
operator module over N;(2) the infima in the expression for the norm in the strong Haagerup
product in the cases discussed in the theorem are actually achieved;

tensor(3) if Z is also a dual operator module, then by (11) and the discussion below
it, an operator space predual of Y®ny
Z is Z, ® mY. Moreover, from thisobservation it is easy to see that the formal sums defining ®;5s converge inthe weak*-topology of Y®nsZ induced by this predual.

(4) If Y, Z are right W*—modules over M, then T' € B(Y, Z) with ||T|| < 1 if, and
only

if,

there

is

an

indexing

set

I

such

that

we

can

write

T = 3kes loe >< yrl, where | ocr
3
lzr >< zelll < 1, and | 32peq lyk ><yk||l < 1. Here we use the Dirac notation |y >< z| for the operator y < z|- >.
All the sums here converge in the weak*-topology (and the last two converging as such implies the first does too). This result, however, is well known,
and follows from standard methods.
Results similar to 3) and 4) hold for left modules.
3

Tensor products of W*—modules

Let M and N be W*-algebras. A (right) M — N correspondence is a right
W*—module Z over N, which is a left M-module via a normal representation
6: M — B(Z). A left M — N correspondence is a left W*—module over M which
admits a normal right action of N. The notion of a correspondence is due toConnes. His definition Connes [94] looks dissimilar to ours (which is what Rieffelcalls a selfdual normal N-rigged M-module in Rieffel [74a]); we are not sure whom
to attribute for the fact that they are equivalent, probably Rieffel [1974a] and Baillet

et al. [1988] (where it is made explicit). It follows as remarked in Section 2 from

David74

P. Blecher

looking at the linking W* —algebra for such a Z, that Z is a strong left module overM, and a strong right module over N. In fact it is easy to see that Z is a 2-sided
dual operator module. If in addition Y is a right W*—module over M then thecomposition tensor product Y®¢Z is defined to be the selfdual completion of theC*—module interior tensor product Y ®¢ Z of Y and Z.
We now show how to use the preceding theory to obtain some results in

Denizeau and Havet [94], and some other results.
Theorem 3.1 Let Y be a right W*—module over a W*—algebra M, let Z bea right W*—module over a W*—algebra N, and let 0 : M — B(Z) be a normal
representation. Then the composition tensor product Y®¢Z is completely isomet-

rically w*-isomorphic to Y®rmZ.

The conjugate W*—module (Y&Z) = Z&rmY

completely w*-isometrically. The predual of Y®¢Z
morphic to Z, Our YeN.ovZouY.
Proof Let Y ®¢ Z be the C*—module
8.2 in Blecher [1995] we haveis completely isometrically iso-interior tensor product.

By Corollary

CBN(Y ®¢ Z,N) = CBy(Y,CBn(Z,N)) = CBu(Y, Z) = Z&uuY
completely isometrically, using Theorem 2.5. Of course By (Y®¢Z, N ) = CBn(Y®¢Z,N). Thus Z&nmY is the selfdual completion of the conjugate C*—module ofY ®¢ Z. Similarly, Y®narZ is the selfdual completion of ¥ ®¢ Z. Unravelling theidentifications above we see that the embedding of Y ®¢ Z in its selfdual completion
corresponds to the natural inclusion of Y ®¢ Z = Y ®nm Z in YRurmZ. The laststatement(1).

of the theorem

follows from Remark

now

3 at the end of Section 2, andO

Since by Remark 3) after Theorem 2.5 the sums defining Y®pmZ converge
in the (unique) weak*-topology of Y®nmZ, and since the inner product on a
W* -module is separately weak*-continuous, we have that the inner product onY®eZ = Y®npZ may be expressed as
< >

yeu)

iel

=; uw;
i€J

=)

>=

SO

< z)0(<

yilz; >)w;

>

ig

"3

<alb(< pile; >)w; >= 27[0(< wile; >)w,

FR
where z = (2;),w = (w;), and the last matrix product is computed in the linking
W*—algebra. The sums here converge in the w*-topology of M. Here I is anarbitrary set, so that the formula above explicitly gives the i.p. for the most generalelements of Y®gZ. Part of the following result may also be found in Denizeau andHavet [1994], Corollaire 1.2.6 and Lemme 2.1.5.
Corollary 3.2 An element u in the composition tensor product Y®¢Z has
norm < 1 if, and only if, u may be written as a sum u = Yker Yk ® 2x, whereye € Vz, € Z satisfy | Sper < zlze > | <1, and | Xges lye >< welll =I< wily; >lajll < 1. All sums here converge in weak *-topology (and the last two
w*-converging imply that the first does too).
The following result is an immediate corollary of Theorem 3.1 and Theorem
2.5. Its main assertion, that the composition of correspondences may be identified
with a space of completely bounded maps, is contained in Denizeau and Havet

On

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HilbertModules

75[1994], Proposition 2.1.1 and Theorem 2.2.2. The first two lines of Theorem 3.1
contains our alternative proof of this result. B. Magajna has recently shown theauthor another route to proving 3.1, and consequently also to this result.
Corollary 3.3

LetY, Z be right W*—modules overM and N respectively, andsuppose that 6 : M — B(Z) is a normal *-homomorphism.
product Y®¢Z = Bp (Y,Z)Y®Z).

Foru€Y®yZ

The composition tensor

completely w*-isometrically (via the canonical map on

write i for the induced map

map if u is an elementary tensor).

¥ — Z (this is the obvious

Then we havelull = lldlles = min{|| fllllgllcs}
where the minimum is over all indexing sets I and (completely) bounded left Mmodule maps f : Y — RY(M), and g : R¥(M) — Z with @ = gf. If, further,
Z 1s also a left W*—module over M, then we have YZ = B M(Z,Y) completely
w*-isometrically.
It is not sufficient in the first result in the corollary above to use the bounded

norm on Bp(Z,Y) as we noted in Blecher [1995].

As a consequence we note that if X is a left W*—module over M ,and Z is aright M — N correspondence, then CBy(X, Z) = B(X, Z) is a right W*—module

over N via the action (T - n)(z) = T(z)n. The inner product here is < S|T >=Yo <S(ea)|T (ea) > where {e,} is ANY o.n.b. for X. In particular, CBu(X, 2)
is the set of module maps T': X — Z with 3", < T(es)|T (eq) > weak*-convergent
in N, and for such T' we have ||T||, = | 3°, < T(€a)|T(€a) > ||z. This appar-

ent generalization of the Hilbert-Schmidt notion was investigated in Denizeau andHavet [1994], and would be an interesting object for further study.
We also obtain a description of the ‘bounded operators’ on a composition tensor
product.
Corollary 3.4 We have B(Y®yZ) = Y®ruB(Z)@nMY completely isometrically w*-isomorphically. The operator space predual of B(Y ®¢Z ) is
Y

®u

Z

On N, SN

Z8uY

2Y

@y

BZ).

ou

Y

completely isometrically.
Proof The first relation is because

B(Y®Z) = (Y®Z)R®nN(Y®0Z)& (YOM

Z)®rN (ZBrmY)

fa Y®nuB(Z)®nMY.

The weak*-isomorphism follows as in Theorem 2.5. To prove the assertion about

the predual, we note that using the first relation in Theorem 3.6 we have:
B(Y®Z,Y®¢Z)= CBum(Y,CBn(Z,,Y®pZ))
2CBu(Y®m Z,YR¢Z) 2 ((Y Qu Z) On (Y®eZ).)*The last part uses (1). Thus from the last part of Theorem 3.1 the predual isY ®um Z) ®n N, ®n Z8uY.
Grouping the three middle terms and using (11) gives the last relation.

[JP. Blecher

David

76

If Z is also a W*—equivalence bimodule (see Rieffel [74a,74b}) for

M , orindeed if 6 is faithful with range B(Z), then it follows immediately from 3.4 that

B(Y®Z) = B(Y) since B(Z) 2 M. This is well known (Denizeau and Havet [1994],
Proposition 1.2.3 (i)). We have been told that the following result is well known

too, however we include it together with its one line proof since we are unable togive a specific reference.

Corollary 3.5 A
T €e B(Y®oZ)

in

Ty

net

boundedconverges

B(Y®Z)

weak*to€ Z we haveif, and only if, for all y,y' €Y, 2,2’<T(y®2)y ®7 >-<T(y®2)|y ®2 >weak* in N.
The necessity here is obvious, the sufficiency follows from Corollary 3.4 since
the hypothesis extends easily to finite rank tensors, which are norm dense in thelast space mentioned in the statement of 3.4.
We observe here too, because we have not seen this in print, both left and right
distributivity of the composition tensor product over arbitrary W*-module direct
sums. This is easy using the universal property 2.2 and the previous corollary.
Next we transfer the HOM - tensor relations in Chapter 20 of Anderson and

Fuller [1992] into our context, with HOM = B(—), the completely bounded modulemaps.
be W*—algebras.

We have the following com-

(1) By(Y®6Z,W) = Buy (Y,BN(Z,W)) if Z is an

M — N correspondence, Y is

Theorem 3.6 Let M and N
pletely isometric identifications:a right W*—moduleN.
:over M,and W

is a strong right operator module over(2) Bp (Y, (Z&rnW)) = ZRpNBum(Y,W) if Y is a right W*—module over M, if
W is a strong N — M operator bimodule and if Z is a strong right N -operator
module.

(3) By(Bum(Y,W),X) 2 Y®umBn(W, X) if Y is a right W*—module over M,
X is a strong left N-operator module, and W(4) Bu (X,Bn(Z,W))
modules over M
tor bimodule.

= BN(Z,Bu(X,W))
and N

respectively,

is a left

if X,Zand if W

N—M correspondence.

are left and right W*—
is a strong M — N

opera-As in Blecher [1995] we remark that these are (right) versions, of the four HOM- tensor relations in Chapter 20 of Anderson and Fuller. In fact (4) is a symmetriccondition, but (1)-(3) have ‘left’ versions whose statements we leave to the reader.
The proofs all follow easily from Theorem 2.5, therefore we shall prove only

one of the relations above, namely (3):By (Buy (Y,W),X)=By(WhnY,X)= YR= (WRrMY)BrNX

WRKrNX = Y ®nuBn(W, X).

One would expect these identifications to be w*-homeomorphic whenever thatmakes sense. We checked all of these except (2), using the ‘canonical preduals’B(Y, Z), = Y ®u Z. for an operator module Y" and a dual operator module Z (seeOn

Selfdual

HilbertModules77

remarks after (11), and Remark 3 after Theorem 2.5). As illustrated by example
in Blecher [1995], these HOM-tensor relations are not generally true without completely bounded maps. As a corollary of (1) above we see that By (Y®pZ, W) =Bn(Y ®um Z,W) completely isometrically, where Y,Z, W are as in (1). We havealready used this relation in Corollary 3.4.
We have not seen the following definition in the literature, although it is theselfdual completion of the construction in Rieffel [74a] Proposition 8.5. Suppose that
Y is a right W*—module over a W*—algebra M, and that Z is a right W*—module
over a W*—algebra N. We define the W*-exterior tensor product Y®Z, whichis to bea W*—module over M®Nlinking W*—algebra for Y (resp.

as follows.

Z).

Let L(Y)(resp.

L(Z)) be theIdentify Y ® Z with the obvious subspaceof the W* algebra tensor product L(Y)®L(Z). Write Y®Z for its completion in
the weak*-topology of L(Y)®L(Z). As we mentioned in (111) in Section 2, wehave Y®Z = (Y, ® Z,)* completely isometrically. Thus we have from canonicaloperator space identifications that Y®Z = CB(Y.,Z) = CB(Z.,Y) completely
isometrically. By viewing Y®Z as in the tensor product of the linking W*—algebras,
we see that it is obviously a M®N module, and has a natural
M®N valued innerproduct.
Its selfduality may be seen in several ways, one of which is given in
Theorem 3.8 below.
Lemma 3.7 Suppose for k = 1,2 that Ty : Yi, — Zx are completely bounded
weak *-continuous maps. Then Ty ® Ty defines a unique completely bounded weak*-continuous map Y18Ys — Z1®Z,. Moreover, ||T1 ® To|lcs < | Th llcbl| T2)|chProof This is straightforward and well known,

T; ® T; may be taken to be(Th) ® (T2)4)*, viewing (T1)« ® (T2)x : (Z1)x ® (Z2)« — (Y1)+ ® (Ya). The latteris completely bounded since ® is functorial (see Blecher and Paulsen [1991]), withc.b. norm < ||(T1)xllesll(T2)sles = IT1llctl| T2llcs. This proves the result.

[J

Theorem 3.8 The W*-exterior tensor product of W*—modules Y and Z is a
W*—module.

Its operator space predual is Y, ® Z, completely isometrically.

If {z,}

is an o.n.b. for Y and if {wg} is an o.n.b. for Z, then {zo ® wg}lap is an o.n.b.
Jor Y®Z .
Proof Let ¢ : Y — CF (M) (resp. ¢' : Z — C¥(N)) be as in Theorem 2.1 forY (resp. for Z), coming from the o.n.b.’s. Note that
CTY (M)RCY(N)= Ci®MRC;®N= Cy; (MAN)completely isometrically. ~~ Via this identification of C¥(M)®CY¥(N) withC¥;(MQ®N), the inner product on C¥(M)QCY¥(N) is easily seen to be given by
the formula<YRzly ®2 >=<yly >®@< 22’ >for y,y € CY(M),z,2' € CY(N).

Thus C¥(M)RCY¥(N)

is a W*—module over

MQ®N,

is a W*—module overM®N using Theorem

from which we see that Y®Z

2.1 and the functoriality of ®, since ¢ @ ¢' : YR®Z — C¥(M)RCY¥(N) embeds
Y®Z in CY (M)®CY(N) as a 1-complemented summand. The statement about
the o.n.b. follows immediately from the explicit embedding just constructed.

[J

78

David

P. Blecher

Since both operator space tensor products are functorial (see 3.7 and 2.3) we
get a quick proof of the following result, most of the statements of which are known:
Corollary 3.9 Suppose that Y1,Z1 are right W*-modules over M, Ys, Zs areright W*-modules over N, and that T; : Y; — Z; are bounded right module maps.
(i) Then Th ® Ty is bounded as a map Y1QYz — YAY OR(ii) If, further, @ and 7 are normal *-homomorphisms from M

to B(Y2)

and

B(Z,) respectively, and if Ts is a left M-module map, then T1 ®T; is bounded
as a map YV1&pYo — 218,25 .

In both cases, |T1 ® Tz|| < | Th||IT2]| . For (ii), we do not need T> to be a right
module map if it is completely bounded.
4

The Eilenberg-Watts

theoremfor W*—modules

In Rieffel [1974a], it was shown that isomorphisms between categories of Hilbert
spaces of normal representations of two W*—algebras are implemented by the composition tensor product with a fixed W*—equivalence bimodule.
Moreover, heshowed that the correspondence between such functorial isomorphisms and modules was bijective. That is, he proved a version of Morita’s fundamental theorem
in the setting of W*—algebras. En route, the Eilenberg-Watts theorem for such
categories was established. An Eilenberg-Watts theorem should state that an additive covariant functor between categories of right modules (over two different
rings) is naturally equivalent to tensoring with a fixed bimodule. In his paper it
is made clear that no such theorem can exist for C*—algebras, without considering extra structure (and at present it is unknown what that extra structure couldbe). In Blecher [1995] we found the appropriate Eilenberg-Watts and fundamental Morita theorems for the category of C*—modules. As a consequence, we wereable to give a functorial significance to the notion of strong Morita equivalence ofC*-algebras, namely that it is indeed a Morita equivalence of some category. Our
methods transfer easily to W*—modules, however here these results seem to be

known although not explicitly written out (see Ghez et al. [1985] - we thank W. L.
Paschke for bringing this reference to our attention). Because of this, and becauseof the lack of difficulty in this case, we omit the proof. We shall merely state theEilenberg-Watts theorem, since it may be viewed as a functorial characterizationof correspondences, complementing Rieffel’s characterization in Rieffel [1974a).
If M is a W*—algebra, let W*MOD), be the category of right W*—modules
over M with morphisms the bounded M-module maps. This is called Hmod(M) in

Ghez et al. [1985], which we refer to for additional terminology. If Z is an M — N
correspondence as in Section 3, then the composition tensor product —®sZ is anormal *-functor from W*MODj; to W*MODy (use Corollary 3.5 and the wellknown fact that a map is weak*-continuous iff its restriction to the unit ball is
weak*-continuous, to get the normality). We want the converse of this.
Theorem 4.1 (Eilenberg-Watts theorem for W*—modules.) Let M and N be
W*—algebras, and let F be a normal *-functor W*MODy — W*MODy.
Then
there is a right W*—module Z over N, and a normal *-homomorphism 0 : M —B(Z), such that F(—) is naturally unitarily isomorphic to the composition tensor
product —®¢Z, and also to Bp ((—),Z). That is, there is a natural isomorphism
between these functors, which implements a unitary isomorphism F(Y) = Y®Z =Buy (Y,Z) for allY

eW*MODy.

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79

Just as in pure algebra, it is an easy exercise to show that the unitary isomorphism classes of normal *-functors W*MOD,; -W*MODy
is in a 1-1 coITespondence with the unitary equivalence classes of M — N-correspondences. Compositionof such functors corresponds to the composition tensor product of the bimodules.
Finally we remark that it can be easily shown that two W*-algebras M and
N are Morita equivalent if, and only if, there is a Hilbert space H such that

M®@B(H)= N®B(H)*-isomorphically.

This is not written down anywhere as

far as the author can tell, although it is known.
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(2] Baillet, M., Denizeau, Y., and Havet, J. F. [1988], Indice dune
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esperance con-[3] Blecher, D. P. [1992a], The standard dual of an operator space, Pacific J. Math.,

153, 15-30.

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Tensor products of operator spaces II, Canad. J. Math.,[5] Blecher, D. P. [1993], Talks at AMS meeting and Operator spaces workshop,
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[8] Blecher, D. P. [1995], A new approach to Hilbert C*— modules, Math. Annalen.,to appear.

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(12] Connes, A. [1994], Noncommutative geometry, Academic Press.
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