A Generalization of Hilbert Modules

Journal of Functional Analysis 2851 journal of functional analysis 136, 365421 (1996) article no. 0034

David P. Blecher*

Department of Mathematics, University of Houston, Houston, Texas 77204-3476

Received May 12, 1994; revised March 23, 1995

We show that there is a natural generalization of the notion of a Hilbert C*-module (also called ``Hilbert module,'' ``inner product module,'' ``rigged module,'' and sometimes ``Hermitian module'' in the literature) to nonselfadjoint operator algebras, and we lay down some foundations for this theory, including direct sums, tensor products, change of rings, and index for subalgebras of operator algebras. These modules in general do not give rise to a Morita equivalence (unlike in the C*-algebra case). 1996 Academic Press, Inc.

Contents

1. Introduction 2. Preliminaries and inductive limit operator spaces. 3. Rigged modules and adjointability. 4. Prerigged modules and direct sums. 5. Equivalent definitions of rigged modules. 6. Induced representations and tensor products of rigged modules. 7. Direct sums, complementability, and K-groups. 8. Countably generated rigged modules and Morita equivalence. 9. Index for subalgebras of operator algebras.

1. Introduction

In what follows we use the term operator algebra for a (not necessarily self-adjoint) (and usually norm closed) subalgebra A of B(H), the algebra of bounded linear operators on a Hilbert space H. We shall also usually assume, and this is an important restriction, that A has a contractive approximate identity (c.a.i.). This of course includes all operator algebras with identity of norm 1.

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A C*-algebra, of course, is an operator algebra which is self-adjoint. Modules over C*-algebras arise naturally and immediately in the form of the Hilbert spaces on which the C*-algebra can be represented. However, this class of modules may be ``generalized'' to what is perhaps the second most important class of modules over a C*-algebra: the Hilbert C*-modules. Unfortunately, there are a few other names for these objects in the literature (``inner product modules,'' ``rigged modules,'' or ``Hermitian modules''), and sometimes the names contradict definitions given elsewhere. These were introduced over general C*-algebras independently and at the same time by Rieffel ([51] is heralded by the research announce- ment [48], where the basic ingredients are presented), Paschke [43], and Takahashi [56] in the early 1970s. They generalize Kaplansky's notion of C*-modules [37] (which is the case when the C*-algebra is commutative). A (right) Hilbert C*-module over a C*-algebra A is a right A-module which possesses an A-valued inner product, that is, a map ( }|}):Y_YÄAsatisfying: (i) (y|y)0 \y # Y; and (y|y)=0 y=0, (ii) for fixed y # Y, (y|})is linear, (iii) (y|z)*=(z| y) (\y, z # Y), (iv) (y|za) =(y|z)a (\y, z # Y, a # A). We usually assume that Y is complete in the norm &(y|y)&1Â2.IfA=C then (iv) is redundant, and then this is the definition of Y being a Hilbert space. If the span of the range of the inner product is dense in A the module is called full; however, we usually do not make this requirement. The theory of Hilbert C*-modules is one of the most important basic techniques in modern operator algebras. It is used in, or has connections to group representations, induced representations of C*-algebras, K-theory, KK-theory, Rieffel's theory of (strong) Morita equivalence of C*-algebras, Connes theory of noncommutative geometry, quantum groups, and index theory (Jones' basic construction). As indicated in [52], a Hilbert C*- module may be seen as a noncommutative vector bundle (where the dimension of fibers may vary)Swan's theorem [55] illustrates this quite clearly. Indeed Hilbert C*-modules may be viewed as the C*-algebraic version of the idea of a projective module in pure algebra. The main idea of this paper (and its companion [12]) is to extend these techniques and constructions to nonselfadjoint operator algebras. We now proceed to outline a first obstruction and its revealing solution. If A is a unital ring and Y is the direct sum of n copies of A viewed as a right A module in the obvious way, then it is trivial that Hom(Y, Y)is ring isomorphic to the ring Mn(A)ofn_nmatrices with entries in A. Here Hom(Y, Y) is the set of right A-module maps from Y to Y.IfAis a unital

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C*-algebra, then Mn(A) is also naturally a C*-algebra in a canonical way.

The ``analytic version'' of the Y above is what we call Cn(A), which may be thought of as the first column of Mn(A) with the inherited norm and obvious right A-action. Of course, Cn(A) is a Hilbert C*-module, and, indeed, it is the basic building block out of which all Hilbert C*-modules may be constructed, but we shall not need this fact here. It is a well-known, probably folklore, result that the natural ring isomorphism between

BA(Y, Y) and Mn(A) discussed above is also isometric. Here BA(Y, Y)is the set of bounded right A-module maps from Y to Y, with norm the usual norm of a bounded map. This relation may be written as K(Y)$Mn(A)in the language of Hilbert C*-modules, and in a sense that can be made precise, this basic identity underpins the theory of Hilbert C*-modules. Note that the above all makes sense for a nonselfadjoint subalgebra A of B(H). That is, Mn(A) is a well-defined operator algebra, viewed as a subalgebra of B(HÄHÄ }}}ÄH). Also, Cn(A), the first column of

n Mn(A), is a right A-module with a distinguished norm. If A has identity (of norm 1) is BA(Y, Y) isometrically isomorphic to Mn(A) via the natural isomorphism? The answer to this basic question is crucial to any proposed extension of the theory of Hilbert C*-modules to nonselfadjoint operator algebras. Unfortunately the answer turns out to be in the negative (we leave it as an exercise for the reader to find a counterexample). However, it is shown here (and in [12]) that CBA(Y, Y)$Mn(A) isometrically isomorphically (in fact, even completely isometrically isomorphically). Here

CBA(Y, Y) is the space of completely bounded right A-module maps from Y to Y, with norm the completely bounded norm of a linear operator (see [44] or below for definitions). To reconcile this statement to the C*-algebra case we observe that if A is a C*-algebra, then the usual norm and completely bounded norm coincide for transformations in BA(Y, Y). In fact for any bounded module map between Hilbert C*-modules (which is of course some kind of multiplier) the usual norm and completely bounded norm coincidethat is, complete boundedness is automatic (see, for example, [11]). L. G. Brown has told us that this latter fact was known to him. We need to describe how a Hilbert C*-module Y may be viewed as a space of operators so that the notion of complete boundedness makes sense. Here are two equivalent ways to see this: (1) view the module as the obvious subspace of the linking C*-algebra [19]; (2) appeal to Ruan's abstract characterization of operator spaces [53], assigning norms to n 1Â2 Mn(Y)by&[yij]&=&[k=1 (yki | ykj)]& . Let us call this the canonical operator space structure associated with the Hilbert C*-module. The converse is true: namely the operator space structure on a Hilbert C*-module determines the inner product (in fact the norm determines the i.p. [11]). Thus for Hilbert C*-module constructions we can forget about the inner

File: 580J 285103 . By:BV . Date:01:02:00 . Time:13:54 LOP8M. V8.0. Page 01:01 Codes: 3514 Signs: 3025 . Length: 45 pic 0 pts, 190 mm 368 DAVID P. BLECHER product and work with the operator space structure, being able at any time to rederive the inner product. We remark that automatic complete boundedness of adjointable maps seems to hold in surprisingly many nonselfadjoint situations too. It would be interesting to characterize this property. In any case, the above shows quite clearly that to easily generalize the C*-theory it is not sufficient in general to use bounded mapswe must (explicitly or implicitly) use complete boundedness and the associated theory of operator spaces. In the light of modern thinking the above discussion is not surprising, and the solution is natural. Arveson showed 25 years ago in [2, 3] (among many other things) the importance of studying matrices over nonselfadjoint operator algebras and their norm and order, and he introduced the study of completely bounded maps. It has become more and more clear that bounded maps are insufficient in many contexts involving spaces and algebras of operators on Hilbert space (this is one of the points of Effros' ``quantized functional analysis'' [27]). In any case, once we accept the idea of using operator spaces and completely bounded maps (which are, as we have seen, essentially forced upon us), the theory does generalize naturally. The class of modules we introduce over a general possibly nonselfadjoint operator algebra does not possess in an obvious way an inner product (although it does in a sense which we will make clear later). For this reason it is perhaps inappropriate to name the generalization ``Hilbert module,'' and therefore we choose to use Rieffel's nomenclature of ``rigged modules.'' The rigged modules over a C*-algebra are precisely the Hilbert C*-modules (see Theorem 5.8), and this observation gives an operator space characterization of Hilbert C*-modules. It seems that our approach provides many insights into the C*-theory, including some simplifications and some new results here [11]. There is one important variation that enters into the nonselfadjoint theory. It is well known in the C*-algebra theory that the study of Hilbert C*-modules is equivalent to the study of (strong) Morita equivalence. That is, every (strong) Morita equivalence gives a Hilbert C*-module implementing the equivalence, and conversely, any Hilbert C*-module Y over a C*-algebra A implements a Morita equivalence between a C*-subalgebra of A and the imprimitivity C*-algebra of Y [52]. This may be merely a fortunate coincidence arising from the fact that every C*-algebra has a contractive approximate identity; at any rate this is far from true in the nonselfadjoint case (see Example 3.3). There is some advantage to us in this: it is consequently easier in general to find examples of rigged modules than to find Morita equivalence bimodules (a Morita equivalence does give rise to a rigged module). We make no effort here to provide many examples, and we concentrate instead on developing the theory.

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The examples which we do give illustrate most of the pertinent parts of the theory. The idea (promulgated by Paul Muhly) is to use the algebraic theory of projective modules, and the theory of Hilbert C*-modules as guides to developing the nonselfadjoint operator algebra scenario. A little thought reveals that it is imperative for this program to have an abstract characterization of operator algebras to free us from dependence on any particular Hilbert space the algebra may be concretely represented on and, also, to ensure that abstract ring constructions actually produce operator algebras. Fortunately an effective characterization is available, namely, the main theorem in [15]. The analysis of [21] leads naturally to an abstract characterization of operator modules (note that in Section 2 of the present paper we give a new abstract characterization up to completely bounded isomorphism of operator modules). Thus one is led ineluctably to study categories of operator modules; and at first glance it appears as if all of basic ring theory can be formulated smoothly in terms of operator algebras and operator modules. However, we run into two obstructions more or less immediately: (1) even if one uses the ``good'' class of morphisms, the completely bounded module maps on a general operator module Y (see our earlier discussion of the necessity of ``complete boundedness''), the endomorphism ring Hom(Y, Y)(=CBA(Y, Y)) is very infrequently an operator algebra and (2) the fundamentally important concept of a direct sum of operator modules is refractory, in general. These obstacles exist of course, in the C*-case, too, but not if we restrict our attention to Hilbert C*-modules. The rigged modules remedy these two difficulties for general nonselfadjoint operator algebras; for such modules Y, Hom(Y, Y)isan operator algebra, and direct sums and other constructions (such as the inner and outer tensor product) can be defined and perform smoothly. We adopt an inductive limit approach to these modules here, which seems to work well. Briefly, Hilbert C*-modules and, more generally, rigged modules over an operator algebra A may be defined as a certain kind of inductive limit of the ``basic'' or ``standard'' modules Cn(A) discussed earlier. This is the content of Definition 3.1. Since Hom(Cn(A), Cn(A)) is an operator algebra, it follows that Hom(Y, Y) is one too. Also, since direct sums of the basic modules Cn(A) are easy to understand and behave well, so will direct sums of rigged modules, which are the inductive limits of these basic building blocks. The reader may at this point be sceptical about the ease of working with an inductive limit definition, particularly in contrast to the simplicity of the definition of Hilbert C*-modules in terms of an inner product. However, there is a host of equivalent definitions (Section 5) which look very similar to an inner product. Essentially (and bear in mind here by way of allegory the distinction between Hilbert and Banach spaces), the A-valued inner

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CBA(Y, Y), namely the closure of the subalgebra YY of ``finite rank operators.'' We show that a rigged module Y over an operator algebra A is algebraically finitely generated and projective if and only if K(Y)is unital. The multiplier algebra of K(Y) is the operator algebra of adjointable operators B(Y) [13]. Also, K(Y) is an inductive limit of finite dimensional full matrix algebras Mn(A). The ``linking algebra'' of 2_2 matrices

a y~ , \ y b + for a # A, y # Y, y~ # Y , b # K ( Y ), turns out to be a nonselfadjoint operator algebra containing all the ``data'' of the rigged module. Conversely, one may define (Theorems 5.4 and 5.5) a rigged module in terms of a linking operator algebra. Most of the above is contained in Section 3. In Section 4 we describe a notion of a ``pre-rigged module,'' and as an example of this we give the first definition of the direct sum of rigged modules. An alternative and more complete development is given in Section 7, together with a criterion for when one rigged module is complemented in another. In Section 5 several alternative definitions of a rigged module are given. In Section 6 we study `ìnduced representations'' somewhat in the spirit of [51]. As in the C*-case, there is an `ìnterior'' and an `èxterior'' tensor product of rigged modules, and it turns out that the `ìnterior'' is simply the module Haagerup tensor product, whilst the `èxterior'' is the operator space spatial tensor product of the underlying operator spaces. These are two familiar operator space constructions (see [14], for instance). Since these coincide for Hilbert C*-modules with the usual constructions [11], this illustrates our conviction that `òperator space'' phenomena were always present in the C*-theory (in this case they are almost invisible, since complete boundedness is automatic). A special case of the interior tensor product is the important ``change of ring'' construction, and using this we define in Sec- tion 7 various K-type or projective class groups and demonstrate their functoriality. We consider a restricted class of finitely generated projective modules to form our K-groups, and various forms of equivalence of these

File: 580J 285106 . By:BV . Date:01:02:00 . Time:13:54 LOP8M. V8.0. Page 01:01 Codes: 3223 Signs: 2757 . Length: 45 pic 0 pts, 190 mm GENERALIZATION OF HILBERT MODULES 371 modules give different sized K-groups, all of which are functorial. The morphisms we consider here are the nondegenerate homomorphisms, also called (S)-morphisms in the C*-theory. In Section 8 we study countably generated rigged modules and, also, connections with the Morita equivalence of operator algebras. Under a natural extra condition on a rigged module Y, A is strongly Morita equivalent to K(Y). Our techniques may be used to give a version of a stable isomorphism theorem in [12] (the case of a quasiunit of norm 1) which, of course, is related to Kasparov's absorbtion theorem, which in turn is related to Swan's theorem. In Section 9 we outline how to extend Jones' basic construction to nonselfadjoint operator algebras. To enable this program we develop in Section 2 some basic theory of inductive limits of operator spaces, modules, and algebras. Let us return to our discussion of the sense in which a rigged module Y over a nonselfadjoint algebra A has an actual inner product. Without loss of generality we may suppose that A is unital (since any rigged module is also a rigged module over the unitization A+). Of course A is contained in, and generates, a unital C*-algebra B. We may view B as a left A-module and a right-rigged module over B, in the obvious way. When one forms the interior tensor product of rigged modules introduced in Section 7, using Y and B, one obtains a rigged module Z over B. This is, of course, none other than the ``change of rings'' or ``induced representation'' process; we have made Y into a rigged module over B. Since B is a C*-algebra, it follows that Z is a genuine Hilbert C*-module over B. That is, Z has an inner product (and the operator space structure on Z coincides with the canonical one determined by this inner product). Notice that this seems to give a new way to produce examples of Hilbert C*-modules. Moreover, it is easy to see that Y is contained completely isometrically in Z via the obvious map. In the same way, the canonical dual module Y is contained completely isometrically in Z , which is simply Z with a conjugate structure. Thus the inner product on Z will restrict to a B-valued inner product on Y, and also on Y . The imprimitivity operator algebra K(Y) is completely isometrically isomorphic to a subalgebra of the imprimitivity C*-algebra K(Z). This is all explained in Theorem 6.8. There is a nice relation between Y and its ``C*-module envelope'' Z (Theorem 5.10). From this one may deduce many facts; for instance, if Y is an equivalence module setting up a strong Morita equivalence in the sense of [12] between A and D, then D $ K(Y) and the C*-algebra C*(A), generated by A in B(H), is strongly Morita equivalent as a C*-algebra to the imprimitivity C*-algebra K(Z) (which contains and is generated by K(Y)). This is Theorem 6.12 (see also [60]). Note that since B is not unique, we will in general get a family of inner products on Y. Nonetheless one would expect that if B is taken to be the

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C*-envelope of A in the sense of Arveson and Hamana [2, 33], then the corresponding Z would be the ``minimal'' Hilbert C*-module envelope of Y in some sense, whereas if B is the universal ``maximal'' C*-algebra containing A, then Z is the `maximal' Hilbert C*-module envelope. Of course, in general, Y is not a Hilbert C*-module over B, since there is no direct B action on Y. We have found from experience that a ``completely bounded view of operator algebra,'' such as is contained herein or in [12, 13], naturally divides into three ``versions'' of essentially the same theory. The first and strongest version is where we consider all morphisms ``norm 1'' throughout. In the C*-algebra theory this is the only version, essentially because all idempotents are similar to self-adjoint projections. For nonselfadjoint operator algebras our view is that the second version is more important: here we allow a 1+= control. To illustrate the difference between this and the first, note that in the disk algebra A(D), in contrast to the continuous functions there, we cannot have fg=1 nontrivially, with f and g having norm 1. However, this is quite possible if we allow the norms to be <(1+=). Fortunately, the theory of this second version (which is essentially the first version pushed into an inductive limit) works out quite prettily, with the ='s not appearing in any of our conclusions. The third version is closest to the purely algebraic version in some ways. Here we take all morphisms to be completely bounded, and we do not care how big the completely bounded constants are, so long as they are uniformly bounded. The theory that developes is mostly parallel to the other versions above, except that everything is ``up to a constant.'' Since we are interested in the isometric theory here we ignore this case. In fact we are concerned mostly with the second version. The first version is contained in here, and surfaces occasionally in concepts such as our ``CCGP'' modules of Section 8. We thank Paul Muhly and Vern Paulsen for innumerable discussions. Indeed some of the ideas for the present paper came from our work with Muhly and Paulsen on Morita equivalence of operator algebras [12]. In Sections 4 and 5 of that paper we give an interesting alternative development (which is closer to the spirit of that paper) of some of the matter introduced here in Section 3, Section 4, and Section 7. Conversely, in Sec- tion 8 we use the techniques of this paper to shed light on some aspects in Section 6 and 7 of [12]. We also thank Terry Loring, L. G. Brown, and Ken Dykema for various discussions, and the referee for his comments and suggestions. At a late stage we added some new material and, in particular, changed our presentation of direct sums (following an idea found in the thesis of Qiyuan Na [42]). Finally, the author wishes to thank the Univer- sity of Missouri at Columbia, and the University of California at Berkeley for visiting positions, during which time some of this work was done.

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2. Preliminaries and Inductive Limit Operator Spaces

Our notation is standardsee [44, 27, 23, 14]. We use letters H and K to denote Hilbert spaces. We always use the letter K to denote the space of compact operators on an infinite dimensional separable Hilbert space. If the Hilbert space is n-dimensional we write Mn for the associated space of operators. A (concrete) operator space is a linear subspace Y of B(H), for some Hilbert space H. A (concrete) operator algebra is a subalgebra A of B(H). A concrete (right A-) operator module Y is a subspace Y of B(H), which is right invariant under multiplication by a subalgebra A of B(H). Three or four theorems give very satisfactory abstract characterizations of these objects, as we see below. We write for the spatial tensor product. If X and Y are operator spaces, contained in B(H) and B(K), respectively, then the spatial tensor product X Y may be regarded as the closure in (HK) of the span B of the elementary tensors xy for x # X/B(H), y # Y/B(K). We write M (Y) for Y M . A linear map T: Y Z of operator spaces is completely n n Ä bounded if the map TId K defined on XK Ä ZK is bounded with respect to the spatial norm . For this to hold it is sufficient [44] that the maps Tn=TIdMn : Mn(Y) Ä Mn(Z) be uniformly bounded, and in this case the smallest uniform bound which works for all n is the norm of

TIdK . We write &T&cb for this bound. We say that T is completely contractive if &T&cb1 and completely isometric,ora complete isometry,if

Tn=TIdMn is an isometry for all n. We say that T is a complete quotient map if T is a complete contraction, and each Tn is a Banach space quotient map (that is, they become isometries if one quotients by their kernel). In the algebra (resp. module) case we usually require that our morphisms to be homomorphisms (resp. module maps). We say that a morphism T is completely bicontinuous,oracb isomorphism if it is an algebraic isomorphism with T and T &1 completely bounded. We usually identify two operator spaces Y and Z if they are completely isometrically isomorphic, that is, if there is an isomorphism of Y onto Z which preserves the norms of matrices (in Mn(Y)). Indeed we forget H as soon as possible and carry the information we need in the pair (Y, [&}&n]), where Y is a vector space, and &}&n is a norm on Mn(Y), for all n. We can do this without fear, by virtue of Ruan's theorem (see [30] for a simplified proof ) which asserts that if Y is a vector space and if there is a sequence of norms [&}&n]n=1, with &}&n a norm on Mn(Y), then Y is completely isometric to a linear subspace of B(H), for some Hilbert space H, if and only if the following two conditions are satisfied:

(i) &: } y } ;&n&:&&y&n&;&, for all :, ; # Mn(C), and y # Mn(Y). x 0 (ii) &[0 y]&n+m=max[&x&n , &y&m], \x # Mn(Y) and y # Mm(Y).

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We note that the spatial tensor product of operator spaces, as an operator space, does not depend on any particular Hilbert space representations of the operator spaces concernedsee [44, 14]. We write CB(Y, Z) for the space of completely bounded maps from Y to Z, with the completely bounded norm. A crucial observation [28, 14, 29] is that

CB(X, Y) is an operator space, via the identification Mn(CB(X, Y))$ CB(X, Mn(Y)) which assigns matrix norms to CB(X, Y). The importance of realizing mapping spaces as operator spaces cannot be overestimated, and nearly all recent progress in operator space theory hinges on this. If X, Y, Z are operator spaces and T: X_Y Ä Z then we say that T is completely bounded (in the sense of [22, 45]) if there exists a constant K n such that &[k=1 T(xik , ykj)]&nK &[xij]&n &[ yij]&n for all n, and all [xij]#Mn(X),[ yij]#Mn(Y). The least K is written as &T&cb and T is completely contractive if &T&cb1. (The underlying tensor product corresponding to this class of bilinear mappings is the very important Haagerup tensor product, introduced by Haagerup in an unpublished manuscript, and subse- quently developed primarily by Effros, Paulsen, Smith, Ruan, Christensen, Sinclair, and the author (in [14, 8, 16, 12])). There are two abstract matrix norm characterizations of operator algebras [15, 9], both based on a powerful result of Christensen, Sinclair, Paulsen, and Smith. The latter characterization is easier to state: an operator space with an algebra multiplication is completely boundedly isomorphic to a concrete operator algebra if and only if the multiplication is completely bounded, namely, if and only if matrix multiplication is uniformly bounded. More precisely, there exists a constant K such that n &[k=1 aik bkj]&nK &[aij]&n &[bij]&n for all n, and [aij], [bij]#Mn(A). The characterization in [15] states that under the conditions above, if A has an identity of norm 1 or a contractive approximate identity (c.a.i.), then a complete isometric isomorphism with a concrete operator algebra is possible in the above if and only if we can take K=1 in the last paragraph. In what follows, when we use the term ``operator algebra'' we shall mean unless stated to the contrary, that A is completely isometrically isomorphic with a concrete operator algebra with c.a.i. However, in Section 7 we will need to apply the characterization to an algebra which has an apparently more general type of ``approximate identity.'' We now show that such can be reduced to the usual type. We shall say that a Banach algebra A has an iterated approximate identity

[e(:, ;)] indexed by two different directed sets, if for each a # A and fixed :, the limits lim; e(:, ;) a and lim; ae(:, ;) exist, and if lim: lim; e(:, ;) a=a and lim: lim; ae(:, ;)=a. The following may be well known.

Lemma 2.1. An iterated approximate identity for a Banach algebra may be reindexed to become an ordinary approximate identity.

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Proof. Suppose that [e(:, ;)] is an iterated approximate identity for a

Banach algebra A. Define for each a # A and fixed :, the quantities x:(a)= lim; e(:, ;) a and y:(a)=lim; ae(:, ;) . We define a new indexing set 1 consisting of the set of all 4-tuples #=(:, ;, V, =), where V is a finite subset of

A and where =>0, such that &e(:, ;) a&x:(a)&<=, and &ae(:, ;)&y:(a)&<= for all a # V. It is not hard to show that this is a directed set with ordering

(:, ;, V, =)(:$, ;$, V$, =$) iff ::$, V/V$, and =$=. We set e#=e(:, ;) if #=(:, ;, V, =).

If =>0 and a # A are given, choose :0 such that ::0 implies that

&x:(a)&a&<= and &y:(a)&a&<=. Choose ;0 such that #0= (:0 , ;0 , [a], =)#1. Now if #=(:, ;, V, =$)#0 then &e# a&a&

&e(:, ;) a&x:(a)&+&x:(a)&a&<=$+=<2=. Thus [e#]# # 1 is a left approximate identity. Similarly it is a right approximate identity. K Suppose that A is a concrete algebra of operators on a Hilbert space H, acting nondegenerately, which has a c.a.i. but no identity. We always unitize A by adjoining the identity operator on H. This endows A+= AÄC with an operator algebra structure which up to completely isometric isomorphism is independent of the Hilbert space H. Explicitly, the matrix norms on A+ are given by:

&[aij+*ij }1]&=lim &[aij e:+*ij } e:]& :

(=sup[&[aij b+*ij } b]& : b # Ball(A)]).

Here [e:] is the c.a.i. We now turn to modules, which are the main subject of this paper. We recall that a Banach (right-) A-module Y is a Banach space and a (right) module over a Banach algebra A, such that the module multiplication is bounded as a bilinear map. We say that the module action is essential,or that Y is an essential A-module, if the span of terms y } a (for a # A, y # Y) is dense in Y. Otherwise, the essential subspace of Y for the action of A is the closure of this span. An important theorem here is the strong form of Cohen's factorization theorem [35], which states that if A has a bounded approximate identity, then Y is essential if and only if any y # Y may be written as a product y$a for some y$#Y, a#A (in fact if &y&<1 then we can take &y$&, &a&<1). For such algebras we may take this to be the definition of essential. Of course it follows that the span of terms y } a (for a # A, y # Y) equals Y, and we use this last criterion to define ``essential'' for an algebraic A-module Y. Let A and B be operator algebras. An abstract cb-A&B-operator module Y is an operator space which is also an A&B-bimodule such that the module actions are completely bounded, namely, if and only if the matrix module multiplications are uniformly bounded.

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Theorem 2.2. Suppose Y is an abstract cb-A&B-operator bimodule. Then there exists a Hilbert space K, and completely bicontinuous maps %, ?, ,ofA,B,and Y, respectively, into B(K) such that % and ? are homomorphisms, and

,(a } y } b)=%(a) ,(y) ?(b) for all a # A, b # B, y # Y. Proof. Consider the algebra D of 2_2 matrices

a y _0 b& for a # A, b # B, y # Y. The product here is the formal product of 2_2 matrices, implemented using the module actions and algebra multiplications. Give the algebra D any operator space structure which retains the original matrix norms on the three nonzero corners (for instance, the structure AÄ YÄ B). It is a simple matter to check that the multiplication is completely bounded. Thus we may appeal to the (first) characterization of operator algebras above [9] to obtain a completely bounded isomorphism \ of D onto some concrete operator subalgebra of B(K), say. Letting

a 0 0 0 0 y %(a)=\ ; ?(b)=\ ; ,(y)=\ , \_0 0&+ \_0 b&+ \_0 0&+ we obtain the desired result. K Thus every abstract cb-operator bimodule is completely boundedly isomorphic to a concrete operator bimodule. The converse is, of course, much easier. The theorem above is a special case of the following result. Suppose that X and Y are cb-A&B- and cb-B&A-operator bimodules, respectively; and suppose that there exist completely bounded bilinear pairings (}, }):X_YÄA, and [ }, } ]: Y_X Ä B, such that the ``linking algebra'' L of 2_2 matrices a y _x b& for a # A, b # B, y # Y, x # X is indeed an algebra with the usual product of 2_2 matrices (using the module actions and bilinear pairings). Then, as in the proof above, we can assign matrix norms and appeal to [9] to obtain a concrete representation of the linking algebra L which is completely bicontinuous on each of the four corners.

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In specific cases we need a completely isometric representation of the linking algebra. This is possible under extra conditions (see [12] and Theorem 5.7 belowsee also Definition 5.4). If Y is an operator space and an A&B-bimodule, if the two module actions on Y are completely contractive and if the algebras A and B have c.a.i.'s, we say that Y is an (abstract) A&B-operator bimodule. Of course, if A=C then we say that Y is a right B-operator module, and similarly for leftmodules. We omit the easy proof of the following.

Lemma 2.3. If Y is an esential operator module over an operator algebra A which has a c.a.i., then Y is also an essential operator module over the unitization A+ (with the obvious action). The C*-algebraic version of the following theorem is in [21], but essentially the same proof works in the nonselfadjoint case. We remark that although the result looks similar to the previous theorem, the proofs are very different.

Theorem 2.4 (Christensen, Effros, and Sinclair [21].) If Y is an abstract A&B-operator bimodule, which is essential with respect to both module actions, then there exists a Hilbert space K, and %, ?, , as in the previous theorem, but now %, ?, , are complete isometries.

If Y and Z are right (cb-)A-operator modules we write CBA(Y, Z) for the space of completely bounded (right) A-module maps from Y to Z.Ifit is necessary to distinguish the right, left, and bi-module cases, we shall specify that at the time. c We recall briefly the standard right A-rigged module H A over a C*-algebra A which may be viewed as a trivial noncommutative vector bundle, or a ``free module'' over A. Authors usually denote this as HA ; however, our notation allows us to distinguish more easily between c r standard rigged right modules H A and left modules A H . Here ``r'' stands c for row, and ``c'' for column. The right module H A consists of certain finite or infinite columns with entries in A, with an obvious inner product [59]. More precisely, given a Hilbert space H, consider the spatial tensor product of K(H) and A. If this is viewed as comprising of large matrices with entries in A (which we may certainly do after fixing an orthonormal c basis for H), then H A consists of the matrices supported on the first column. The matrix may in fact be viewed as a column vector x with entries in A, and then the inner product is simply (y|z)=y*z. ThereÄ is wc Ä Ä Ä Ä another important related construction, H A , where ``wc'' is short for ``weak wrong 2 column'' [12] (Niels Wegge-Olsen calls this module ``H A '' [59]). This may be defined by realizing A/B(K), regarding B(HK) as large matrices with entries in B(K) and by considering the subspace of these

File: 580J 285113 . By:BV . Date:01:02:00 . Time:13:54 LOP8M. V8.0. Page 01:01 Codes: 3255 Signs: 2708 . Length: 45 pic 0 pts, 190 mm 378 DAVID P. BLECHER matrices with entries from A in the first column and zeroes elsewhere. wc Certainly H A is not in general a rigged module, but it is rather an extension of the idea of a rigged module which is very useful, in particular if one wishes to consider W*-versions of the theory. The ``left'' module version of wr wc this is A H . We may also view an element y of H A as a column vector t wr Ä with entries in A, and an element x # A H as a row vector. One can show that the ``finite sub-columns'' of y Ä are uniformly bounded in norm, and in wc Ä c fact this characterizes H A . Of course, H A is completely isometrically con- wc tained in H A , and in fact it is the subspace consisting of the elements y for which the finite ``subcolumns'' form a Cauchy net and, hence, convergeÄ in wc c the norm. If H is finite dimensional then H A and H A coincide, and we c 2 write Cn(A) for the rigged module H A when H=l n (see [14]). If H is the infinite dimensional separable Hilbert space (with a prescribed basis) we c w wc write C (A) for H A , and C (A) for H A , with obvious modifications in the left-module case. And finally we note that all the above makes sense for A a (not-necessarily self-adjoint) operator algebra, or indeed for A merely an operator space. This all leads us to an important fact about these spaces, which we use several times here. Suppose that X, Y, and Z are complete operator spaces and that ( }, } ) is a completely bounded bilinear pairing X_Y Ä Z.If t r wc t x#XH,y#HY , then the formal matrix product x y of the row and columnÄ (whereÄ we use the bilinear pairing to multiplyÄ Ä corresponding entries) actually converges in norm to an element of Z. This is easily seen by examining the norms of products of finite subrows and columns and by wr showing that we have a Cauchy net. A similar assertion holds for X H and c H Y . If A=C, then we write the associated standard right rigged modules as c H , C ,orCn. These are also known as the column Hilbert space and r similarly for H , R , Rn , the row Hilbert spaces. As pointed out in [12] and elsewhere, H c =H c Y=H c Y. Here Y h wc h is the Haagerup tensor product [45]. One can similarly realize H Y as a weak Haagerup tensor product.

Let us write Dn(A) for the subalgebra AIn of Mn(A)(1n ). Of course A$Dn(A). We note that Cn(A) is a left Dn(A)-operator module, and we say that a right A-module map : Cn(A) Ä W is right A-essential if the induced module action (a)( } )=((aIn) } ) is essential. That is,

e: Ä for a bounded approximate identity e: of A, or equivalently by Cohen's theorem, that =$a for some $: Cn(A) Ä W, and a # A. This implies that (and is equivalent to, if A acts essentially on W) is left multiplication by a certain fixed element of Rn(W)ifn< . It is also equivalent, if A acts essentially on W, to being able to extend to a module map on + + Cn(A ), where A is any operator algebra unitization of A (if A is not already unital). In the case n= , let us say that : C (A) Ä W is finitely

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A-essential if the restrictions of to Cn(A) are right A-essential. We shall see in Proposition 2.5 that this implies that (and is equivalent to, if A acts w essentially on W) is left multiplication by a fixed element of R (W), which is also equivalent to having a completely bounded extension to + C (A ). The second half of the next proposition we shall not need until later.

Proposition 2.5. If Y is a right A-operator module, then for n # N

(i) CBA(Y, Cn(A))$Cn(CBA(Y, A)) completely isometrically, and

(ii) the right A-essential part of CBA(Cn(A), Y) is completely isometrically isomorphic to a submodule of Rn(Y), with equality if A acts essentially on Y.

Also,

w (i$) CBA(Y, C (A))/C (CBA(Y, A)) completely isometrically, and

(ii$) The right finitely A-essential part of CBA(C (A), Y) is com- w pletely isometrically isomorphic to a submodule of R (Y), with equality if A acts essentially on Y. If A acts essentially on Y then these are the A-module + + maps which extend to a right A -module map on C (A ) with no increase in cb-norm.

Proof. The first statement follows immediately from the definition of the matrix norms on CBA(Y, A), and is true more generally if one replaces

Cn with Mmn . To see the second statement, notice that an A-right-essential module t n map F # CB(Cn(A), Y) may be written as F([a1, ..., an] )=k=1 ykak, for some elements y1 , ..., yn # Y. An easy calculation shows that

&F&cb&[ y1 , ..., yn]&. However, consider evaluating F on the matrix E;=[E1, ..., En], where Ek is a vector in Cn(A) with e; as the kth entry and zeros elsewhere. We see that &F&cb&[ y1 e; , ..., yn e;]&. Taking the limit gives &F&cb=&[ y1 , ..., yn]&. There is a similar argument at the matrix norm level.

Let us prove (i$). If f #CBA(Y,C (A)) and if ?k is the projection of

C (A) onto its kth coordinate, then set fk=?k b f. The finite ``subcolumns'' t of [ f1 f2 }}}] are uniformly bounded, by the argument in (i) above, and so t w it follows that [ f1 f2 }}}] #C (CBA(Y, A)). Using (i) and considering the uniform bound mentioned above gives the isometry in (i$). To obtain the complete isometry we can either use a similar argument, or deduce it from the isometry. The proof of (ii$) is similar to (ii), and we omit the details. K

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Although we make no essential use of it until much later, we will also refer occasionally to the module Haagerup tensor product hA see [12 or 11] for more details. The most important properties of this tensor product for us are that it is associative and that it linearizes completely bounded ``balanced'' bilinear maps. Also, we will sometimes work with multiplier algebras of operator algebras, for which we refer the reader to [47] and a forthcoming paper of ours [13]. The use we make of multiplier algebras here will be fairly elementary for anyone familiar with the C*-theory. For the remainder of this section we will discuss a certain kind of inductive limit of operator spaces. As an aside we note that the morphisms we work with are completely contractive, which makes things a little simpler. If we work with general completely bounded morphisms, the arguments and statements need to be modified somewhat. For instance any ``family of completely contractive morphisms'' needs to be replaced by a family of completely bounded morphisms whose cb norms have a uniform upper bound.

Let [Y;] be a family of operator spaces (resp. operator modules, operator algebras), indexed by ; in a directed set 2. Let Y0 be a fixed vector space (resp. module, algebra). Suppose for all ; we have linear maps (resp. module maps, algebra homomorphisms) ,; : Y0 Ä Y; and ; : Y; Ä Y0 , satisfying the following list of conditions:

(1) f;, #=,; b# is a complete contraction from Y# Ä Y; , for all ;, #.

(2) sup; &,;( y)&< for all y # Y0 .

(3) for each y # Y0 , ;(,;( y))Äy in the initial uniform topology on Y0 . Initial uniform convergence (see Bourbaki) means the following: a net

[y;] of elements of Y0 converges to an element y initial uniform if sup# &,#( y;&y)& Ä 0. (Note that a zero net can have a nonzero limit.)

One may assign matrix seminorms to Y0 by the formula &[ yij]&n= sup; &[,;( yij)]& and quotient Y0 by the nullspace of &}&1 . We write Y0 again for this quotient. Using (3) and the triangle inequality, it may be seen that this supremum is also the limit lim; &[,;( yij)]&.

In this way metric properties of Y0 are transferred from the Y; , inducing local structure on Y0 . For instance, it is easy to verify using Ruan's condition [53] (in the algebra case use [15, 9]) that with these matrix norms Y0 is an operator space (resp. operator module, operator algebra). We call this the initial uniform or inductive limit operator space structure. Write Y for the completion of Y0 in this norm. In the algebra case if one wishes to apply [15] instead of [9] one also needs to also give a condition which will ensure that we have a c.a.i. [e:]

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for the algebra. Namely we require that e: y Ä y and ye: Ä y (initial uniform) for all y # Y. It is not necessary in the algebra case, in order to show that Y is an operator algebra, that ; , ,; be algebra homomorphisms, but only that the ,; be an asymptotic homomorphism, or even that it merely satisfy

sup &(,;)n ( yy$)&nsup &(,;)n ( y)& sup &(,;)n ( y$)&, ; ; ; for all n and y, y$#Mn(Y0). This is all that is necessary in order to apply

[15, 9], because the calculation is transferred to the spaces Y; , which in this case we are assuming to be operator algebras and which consequently satisfy the norm submultiplicativity condition of [15, 9]. In the cases we encounter later, the following stronger condition holds:

sup &(,;)n ( yy$)&n=sup &(,;)n ( y)(,;)n ( y$)&. ; ;

Similarly, ; , ,; need not be module maps in the module case, providing we have an appropriate asymptotic condition.

We continue to write ; , ,; for the induced maps Y; Ä Y and Y Ä Y; .

These still satisfy conditions (1)(3), but in addition, ; , ,; are complete contractions. We write (Y, [&}&n])= ([Y;], [;], [;]), or simply

Y= Y; , and say that Y is an inductive limit of the Y; . For convenience later we write E; for the map ; ,; on Y. Recall that f;, #=,; # .

Proposition 2.6. The operator space (resp. operator module, operator algebra) Y; is the unique (complete) operator space (resp. operator module, operator algebra) Y with the following universal property: There are completely contractive morphisms @; : Y; Ä Y , the span of whose ranges are dense in Y, such that @; f;, # Ä @# strongly (for all #); and if g; : Y; Ä W are completely contractive morphisms satisfying the compatability condition g; f;, # Ä g# strongly (for all #), then there exists a unique completely contractive morphism g: Y Ä W such that g@;=g; for all ;. Proof. The uniqueness follows the standard route. To prove the existence, we take @;=; .If[g;] is as above, we may define g(#(x))=g#(x) for x # Y# . We note that g#(x)=lim; g; f;, #(x)=lim; g; ,; #(x), which shows that g is well defined and, also, that g is completely contractive. K

We note that the proof above shows that Y= Y; is an inductive limit of certain of its subspaces ;(Y;). Thus the spaces Y; are inducing a ``local structure'' on Y. In the ``cb-version'' of the theory, we get uniqueness only up to completely bounded isomorphism in the theorem above.

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Theorem 2.7. Suppose that for all ;, Y; is a right A-operator module such that CBA(Y;) is completely isometrically isomorphic to an operator algebra. Then CBA( Y;) is also completely isometrically isomorphic to an operator algebra.

Proof. Let Y= Y; . We define maps 8; :CBA(Y)ÄCBA(Y;) and 9; :CBA(Y;)ÄCBA(Y)by9;(T)=;T,; and 8;(S)=,;S; . These are completely contractive, and 9; 8;(T) Ä T in the point norm topology. It follows from this that the condition of [15] may be verified locally: the calculation is transferred to the spaces CBA(Y;). By simple triangle inequality arguments it follows that for (an n_n matrix of ) operators T from CBA(Y), we have &T&=lim; &8;(T)&=lim; &9; 8;(T)&. For convenience we have omitted to write the subscripts .n on some of the terms here (in the n_n matrix case). Also, &T1 T2&=lim; &E; T1 T2 E;&= lim; &E; T1 E; T2 E;&=lim; &8;(T1) 8;(T2)&. Again the subscripts .n are omitted; remember that in general we are multiplying matrices T1 , T2 of operators here. From this type of equality it is easy to check the multi- plicative norm condition from [15]. K

3. Rigged Modules and Adjointability

We begin by giving the inductive limit definition for a rigged module and note that Hilbert C*-modules satisfy this definition. It is useful for us to view a Hilbert C*-module as a module which is asymptotically generated by the trivial bundles. Let us make this connection a little more precise (see also Theorem 3.1 in [11]). Let A be a C*-algebra, and let Y be a right A-Hilbert C*-module. Let - [e;] be a c.a.i. for the imprivitivity C* algebra K(Y), which we may take n(;) ; ; by a now standard trick [18] to be of form e;=k=1 |yi )( yi |. Here we are using bra-ket notation: the symbol |x)( y| represents the operator z [ x( y | z) on Y. Define maps ,; : Y Ä Cn(;)(A) and ; : Cn(;)(A) Ä Y ; ; t t n(;) ; by ,;(z)=[(y1,z), ..., (yn(;),z)] and ;([a1, ..., an] )=k=1 ykak.It may be checked that ,; and ; are completely contractive. Moreover,

; ,; Ä IdY strongly on Y. We also point out and leave it to the reader to check that for any #, ,# ; ,; Ä ,# in norm. The converse is also true as we shall see in Section 5 (see also Theorem 3.1 in [11]), namely that the existence of such an asymptotic factorization of the identity map is a characterization of Hilbert C*-modules. Before we state the general definition we shall introduce a little notation for right operator module maps Y Ä Y. First, such a map T induces a dual map T*: HomA(Y, A) Ä HomA(Y, A). Moreover, if T is completely contractive then so is T*onCBA(Y,A). For traditional reasons we write the

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For f #CBA(Y,A) and y # Y there is an associated ``finite rank'' operator

Ty, f (sometimes written as yf or | y)( f |) given by Ty, f ( y$)=yf( y$). In this case &Ty, f &cb&y&&f&cb . The proofs of these last two allegations are standard and will be omitted. We refer the reader to the discussion before Proposition 2.5 in Section 2 for the definition of right A-essential maps. K

Definition 3.1. Suppose A is an operator algebra, suppose that Y is a right A-operator module, and suppose that there is a net of positive integers n(;) and right A-module maps ,; : Y Ä Cn(;)(A) and ; : Cn(;)(A) Ä Y such that

(1) ,; and ; are completely contractive;

(2) ;,; ÄIdY strongly on Y;

(3) the maps ; are right A-essential;

(4) For all #, ,# ; ,; Ä ,# uniformly (in norm).

In this case we say that Y is a right A-rigged module. We write E; for the map ; ,; on Y. Let Y =[f#CBA(Y,A):(E;)* f Ä f uniformly], and let

K(Y) be the closure in CBA(Y, Y) of the set of finite rank operators Ty, f for y # Y, f # Y . We shall see later that K(Y) and Y are actually independent of the particular directed set and nets ,; , ; . From the discussion at the beginning of this section we see that Hilbert C*-modules are rigged modules. Before we consider some more examples we make some important remarks. It is clear from the definition that a rigged module Y over A is also rigged over the unitization A+ (and indeed over any closed subalgebra of the multiplier algebra of A containing A). This is because an A-essential map has a canonical extension to the associated module over A+ (or M(A)). We remark that one consequence of conditions (2) and (3) is that a right A-rigged module is automatically essential with respect to the A-action. Condition (4) above is important, and at least intuitively it may be phrased as requiring the dual coordinate charts to be asymptotically compatible. Condition (3) is redundant if A is unital, whereas we shall see that (4) is redundant if Y is algebraically finitely generated and projective.

Note that CBA(Y, A) and Y are left A-operator modules with respect to the natural action (a } f )(y)=af( y). Using the definition of Y one may check that it is an essential A-module.

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In fact we do not care in Definition 3.1 if the maps are completely contractive, have cb norm <1, or may be controlled by 1+=. A rescaling and reindexing of the net is always possible to ensure that the cb norm is <1. From a categorical viewpoint, a more appropriate name for a module possessing the properties described in Definition 3.1 would be column generated and approximately projectivesee Proposition 3.4 and the comments below it. We plan to develop the categorical properties of these modules at a later time.

Example 3.2. The right rigged modules over the complex numbers C are exactly the column Hilbert operator spaces. Indeed from the factorizations through column Hilbert spaces Cn(;) it is easy to see that Y satisfies the parallelogram identity and, also, the matrix form of this which characterizes column Hilbert operator space. See [12] and [11, Lemma 2.2] for more on this. For other operator algebras the situation is much less trivial.

Example 3.3. Let A be an operator algebra with c.a.i., and let p be a 2 projection ( p =p) of norm 1 in Mn(A). Let Y=pCn(A). Then obviously

Y satisfies the conditions of 3.1, factoring through Cn(A) with ,( y)=y, (y)=py. It is easily seen that Y $Rn(A) p and K(Y)$pMn(A) p completely isometrically. In fact, we can show (but omit the proof ) that these are the only examples of finite dimensional rigged modules over a finite dimensional operator algebra with c.a.i. (and hence with identity). For a concrete example of this take A to be the upper triangular m_m matrices, and set p=E11 (see also Example 5.6). Here Y and K(Y) turn out to be copies of C. Thus it is clear that, unlike in the C*-algebra case, Y need not have the same dimension as Y and that there is no associated Morita equivalence. Another example: let E be a closed subspace of B(H) and let

A be the subalgebra of M2(B(H)) of elements with E in the 12 corner, 0 in the 21 corner, and scalar multiples of IH in the other two corners. Then Y=pA is a rigged module over A, where p # A has 1 in the 11 corner and zeroes elsewhere. We note that Proposition 2.5 and condition (3) of Definition 3.1 ensure that each ; is actually left multiplication by a row of elements ; ; ; [ y1 , ..., yn(;) ]ofY, which row has norm 1. Similarly, ,;( y)= ; ; t ; ; t [f 1(y), ..., f n(;)( y)] , where [ f 1 , ..., f n(;)] is a column in Mn(;), 1(CBA(Y, A)), with norm 1. We warn the reader that we will use this notation later and ; ; will refer without comment to elements yk and f k (the latter will also be ; written as xk). This observation gives the equivalence of modules satisfying the conditions of Definition 3.1 with the ``(P)-modules'' discussed in [12]. ; ; The elements yk and f k may be viewed as giving local coordinates.

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As mentioned earlier, rigged modules are ``approximately projective,'' a term which is partially explained by the following proposition (which was also found by Vern Paulsen).

Proposition 3.4. Let Y be a right A-rigged module. Suppose that E is a right A-operator module which is a complete quotient of a right A-operator module F, and suppose that T: Y Ä E is a completely contractive right

A-module map. Then there exists a net [T;] of completely contractive module maps from Y into F such that q b T; Ä T strongly on Y. Here q: F Ä E is the quotient map. Further, if Y is algebraically finitely generated and projective over A and if =>0 is given, then there exists a map T= : Y Ä F such that q b T==T, and &T=&cb<1+=. Proof. We use the notation of Definition 3.1, except that we rescale and reindex the nets ,; , ; to have cb. norm <1. Note that

T b ; : Cn(;)(A) Ä E is right A-essential. By Proposition 2.5, T b ; may be regarded as a finite row [e1 , ..., en(;)] with entries in E and norm <1. Since E is a complete quotient of F, we can find a finite row [ f1 , ..., fn(;)], with entries in F of norm <1 with q( fk)=ek for all k. Again by Proposition 2.5 this row may be viewed as a map in CBA(Cn(;)(A), F). Composing this map with the map ,; gives a map T; : Y Ä F, with q b T;=T b ; b ,; , which of course converges strongly to T on Y. Strictly speaking we cannot finish the proof of the last assertion until after Theorem 3.6; however we proceed as above, using , and maps coming from a decomposition

IdY== b ,= as in Theorem 3.6(6). K In connection with the next proposition it is worthwhile to remark that # # condition (4) of Definition 3.1 is equivalent to requiring that E*; f k Ä f k for all # and corresponding k.

Proposition 3.5. The maps E; lie in K(Y). Also Y is generated by the ; ; elements yk , and Y is generated by the elements f k described above. ; Proof. The first assertion is simply the statement that the f k lie in Y , which follows from condition (4) of Definition 3.1. The second assertion follows from condition (2) of the same definition, the third assertion follows from the definition of Y . K If Y satisfies the conditions of Definition 3.1 (that is, if it is a right A-rigged module) then Y is an inductive limit in the sense of Section 2 of the spaces Cn(;)(A). More precisely, letting f;, #=,; # we see that ; f;, # Ä # strongly (for all #). Thus Y= Cn(;)(A), and has the universal property mentioned in Section 2. It also follows that Y is the inductive limit of its submodules ;(Cn(;)(A)). These submodules may be viewed as the local coordinate patches.

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Condition (4) ensures that Y is an inductive limit of the spaces Rn(;)(A).

Namely, consider the ``dual maps'' , ; : Rn(;)(A) Ä Y :[a1, ..., an(;)] [ ; ; ; i ai f i and : Y Ä Rn(;)(A): x [ [(x, y1), ..., (x, yn(;))]. Then it is easy to see that X=Y satisfies the ``left'' or ``row'' version of Definition 3.1. Indeed the analogue of condition (2) is that ,; ; f=E ;* f Ä f, which follows from the definition of Y . The analogue of condition (4) follows quite easily from condition (2) of Definition 3.1. Thus X=Y is a left A-rigged module and may be viewed as being ``row generated and approximately projective.'' Consequently, we may form the dual module X =Y of X. Fortunately this turns out to be Y again, as we shall see in a minute.

In what follows we often write ( }, } ) or ( }, } )A for the natural pairing (evaluation) X_Y Ä A. It is important to observe here that (X, Y,(},}))is a duality pair. That is, the maps x Ä (x, } ) and y Ä (},y) are completely isometric module maps of X and Y into CBA(Y, A) and CBA(X, A), respectively. It is easy to see the first of these. To see the second we observe that the second map is certainly completely contractive; however recall that Y is an inductive limit of the spaces Cn(;)(A). This means that Y is ``normed'' by the maps ,; , which may be regarded as row matrices with entries in X, and norm 1. This shows the second map is completely isometric. To see that X =Y note that by Proposition 3.5, X is generated inside ; CBA(X, A) by the (images of the) elements yk . Thus by the previous paragraph it follows that X $Y completely isometrically. From all of the above we see that X and Y play exactly symmetrical roles. We note that the natural pairing (evaluation) between X and Y may be written as (x, y)=lim;( ;(x), ,;( y));, where ( }, } ); is the natural pairing Rn(;)(A)_Cn(;)(A) ÄA. It follows immediately from this that the evaluation map X_Y Ä A is completely contractive in the sense of [22, 45]. Except for most of (6), all of the following theorem is also presented in [12], with some variations in proof.

Theorem 3.6. If Y is a right A-rigged module then

(1) CBA(Y, Y) is (completely isometrically isomorphic to) an operator algebra.

(2) The endomorphism algebra K(Y) is a left ideal in CBA(Y, Y), with two sided c.a.i. [E;]. In fact, K(Y)=[T #CBA(Y,Y):TE; Ä T].

(3) Y equals the essential subspace of CBA(Y, A) for the natural right module action of K(Y) on CBA(Y, A). (4) K(Y) and Y are independent of the actual choice of factorization nets ,; , ; .

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(5) Y and Y are essential (left and right, respectively) K(Y)-operator modules (and essential (right and left, respectively) A-operator modules). (6) Y is algebraically finitely generated and projective if and only

K(Y) is unital if and only K(Y)=CBA(Y, Y) if and only if K(Y) contains all right A-module maps from Y to Y if and only if given =>0 there exists some positive integer m and completely bounded module maps ,: Y Ä Cm(A) and : Cm(A) Ä Y, with right A-essential, with &,&cb and &&cb less than 1+=, and such that b ,=IdY . In this case, Y =CBA(Y, A), and indeed Y contains all right A-module maps from Y to A.

Proof. To prove (1) we note that we gave a direct calculation of this in [12]. Alternatively we may appeal to Theorem 2.7 of Section 2. To apply this theorem we need to know that for any n,CBA(Cn(A)) is completely isometrically isomorphic to an operator algebra. For A unital this is fairly obviously just Mn(A) again, using Proposition 2.5 if you like. For general A we note that in [13] we explicitly and quite easily realize CBA(Cn(A)) as the operator algebra of left multipliers of Mn(A). For (2) we note that since STy, f =TSy, f it follows that K(Y) is a left ideal in CBA(Y, Y). Notice also that for y # Y, f # Y we have E T =T and T E =T . ; y, f E; y, f y, f ; y, E*; f From Definition 3.1 and the fact that &Ty, f &cb&y&&f&cb we see these both converge to Ty, f . Thus E; is a c.a.i. for K(Y). Item (3) is more or less immediate from the definition of an essential subspace and the fact that E; is an approximate identity for K(Y).

For (4) we observe that if we have another factorization for IdY and if Y $ is the associated module dual and K$(Y) is the associated algebra, then by (2) we have K(Y) K$(Y)/K$(Y) and K$(Y) K(Y)/K(Y). Moreover, since the corresponding maps E; and E$;$ both converge strongly to the identity map on Y, the proof of (2) above, in conjunction with Cohen's factorization theorem [35], shows that the set inclusions are actually equalities. Thus by a result of Paul Muhly (Theorem 4.17 in [12]), K(Y)=K$(Y). It then follows from (3) that Y $=Y . Item (5) is evident from previous comments. To see (6) we observe that if K(Y) has an identity then by (5), this identity is IdY , which shows using (2) that K(Y)=CBA(Y) and using (3) that Y =CBA(Y, A).

Hence from (2) again, given =>0, there exists ; with &E;&IdY&<=. Since the group of invertibles in K(Y) is open, E; is invertible, with inverse

S, say. Note that we can assume here that &S&<1+=. Then IdY=SE; . Set ,==,; , ==S;. This is the desired factorization of IdY. The reader will easily check that Y is algebraically finitely generated and algebraically projective over A (the latter in any reasonable sense such as that in Hungerfords algebra text). The existence of the above factorization easily

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A-module map p: Cn(A) Ä Y. Clearly we can ensure that p is completely bounded. Since Y is projective, there is a lifting j: Y Ä Cn(A) with pj=IdY.

In fact we can replace these maps (and the factorization maps ; , ,; + + for Y) with right A -module maps factoring through Cn(A ), so let us assume that A is unital in what follows (since K(Y) is the same whether we regard Y as rigged over A or rigged over A+, our assumption is justified).

Let E=jp, this is a projection Cn(A) Ä Cn(A) with range Z, say, and also Z=Ran j. Let _ be the restriction of p to Z, this is the canonical isomorphism Z Ä Y. Since p is continuous, so is _, and consequently by the open mapping theorem, so is j. Hence T;=j; ,; _ converges strongly to the identity map on Z. Thus T; Ä IdZ uniformly, since Z is a finitely generated submodule of Cn(A). Hence there exists ;0 with T;0 invertible in

BA(Z, Z); suppose D is the inverse. Then j=D( j;0) ,;0 . Notice that D and j;0 are A-module maps (or restrictions of A-module maps) between spaces of type Cn(A) and are consequently automatically completely bounded. Hence j is completely bounded. By Proposition 2.5 it follows that n the identity map in CBA(Y, Y) may be written as k=1 Tyi, fi , for some n n yi # Y, fi #CBA(Y,A). Hence E;=E; k=1 Tyi, fi=k=1 TE; yi, fi . This con- n verges to k=1 Tyi, fi=IdY. Thus IdY is in K(Y) since the latter is closed. K Theorem 3.6 part (6) shows that if Y is algebraically finitely generated and algebraically projective (or equivalently, if K(Y) is unital) then for any

=>0, we can factor IdY== ,= through a space Cn(=)(A), where the maps

,= , = have cb norm <1+=. That is, we can replace the convergence in Definition 3.1 by equality (condition (4) becoming redundant), at the expense of moving to a 1+= norm control. This observation completes the proof of Proposition 3.4 (which informs us that Y is projective in a stronger sense than the purely algebraic). Some of the following material shall be in [13]. Let T be a completely bounded map from Y to Z, where Y and Z are right A-rigged modules. We say that T is adjointable if there exists a map T : Z Ä Y such that

(w, Ty)=(T (w), y) for all w # Z , y # Y.

If Y is a finitely generated projective rigged module then it follows from Theorem 3.6(6) that every right A-module map from Y to Z is completely

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As an (important) example, note that the maps ,; and ; of Definition 3.1 are adjointable, and their adjoints are the maps , ; and ; defined before Theorem 3.6. It follows from the fact that (Y , Y) is a duality pair that if a map T is adjointable then its adjoint is unique and also adjointable, with T =T.In this case, both T and T are automatically module maps, T is also completely bounded and &T &cb=&T&cb. Also if both S and T are adjointable then so is ST, and (ST)t=TS.

Corollary 3.7. If ,: Y Ä Z is a surjective completely isometric module map between rigged modules then , is adjointable. Also , : Z Ä Y is a completely isometric surjection, and , is unitary by which we mean that (, &1(x), ,( y))=(x, y) for all x # Y , y # Y. Proof. The factorization nets for Z can be transferred via , to maps for Y satisfying Definition 3.1, and it follows from (4) of Theorem 3.6 and the definition of Y that Z b ,/Y . K We define B(Y) to be the space of completely bounded adjointable module maps on Y.IfK(Y) is unital then K(Y)=B(Y)=CBA(Y), and these include all the right A-module maps from Y to Y. More generally, we define B(Y, Z) to be the space of completely bounded adjointable maps from Y to Z. These spaces are complete as in the C*-case, with the same proof. In [12] B(Y) is called End(Y); the following theorem is in [13].

Theorem 3.8. If Y is a right A-rigged module, then B(Y) is a unital operator algebra containing K(Y) as an essential closed two sided ideal. Moreover, B(Y)$M(K(Y)), the multiplier algebra of K(Y), completely isometrically isomorphically. In [12] we showed that:

K(C (A))=M (A), K(C (A))=A , n n K and more generally, for a right A-rigged module Y,

K(C (Y))=M (K(Y)), K(C (Y))=K(Y) . n n K Indeed these follow fairly easily from Definition 5.4, although we give a method of proof at the end of this section.

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We obtain

B(C (A))=M ( (A)), B(C (A))= (A ), n n M M K and for Y as above,

B(Cn(Y))=Mn(B(Y)), B(C (Y))=M(K(Y)K).

All these identities are true ``completely isometrically isomorphically.'' The ``nonsquare matrix'' versions of the assertions above are true, too, as may be seen by applying the techniques below. It is very useful for many purposes to know that K(Y) is completely isometrically isomorphic to the ``Haagerup module tensor product''

YhA Y of [12]. This observation may be generalized slightly.

Definition 3.9. If Y and Z are right A-rigged modules, then we define the space of compact adjointables K(Y, Z) to be the closure in CBA(Y, Z) of the span of rank 1 operators |z)(x| (which are the operators y [ z(x, y)).

As an example note that the maps ,; and ; are compact adjointables.

Theorem 3.10. If Y and Z are right A-rigged modules then K(Y, Z) is contained in B(Y, Z). Also,

K(Y, Z)$ZhA Y completely isometrically isomorphically

In particular, K(Y, Z) is the set of operators T: Y Ä Z which factor as a product gf, where f: Y Ä C (A) and g: C (A) Ä Z are completely bounded compact adjointables. Moreover, the c.b. norm of such an operator T is the infimum of the products & f &cb &g&cb of the cb norms of these compact adjointables. Also, K(A, Y)$Y and K(Y, A)=Y completely isometrically. Proof. It is easy to show that every ``finite rank'' map is adjointable, and so K(Y, Z) is contained in B(Y, Z). By a standard matrix norm type calculation (see [12, Lemma 4.5], for instance), the map defined on rank

1 tensors by zx [ |z)(x| extends to a complete contraction L of ZhA Y into CBA(Y, Z). However, suppose that T # K(Y, Z) and suppose that n(;) ; ; T=L(u) for some u # ZA Y . Then u;=k=1 T(yk)A xk is an element of ZhA Y and u; Ä u. Since &u;&&T&cb the map L is isometric. A similar argument shows the complete isometry. If u # ZhA Y , &u&<1, then we may write u=k=1 |zk)(xk |, where the sum is norm convergent, and, indeed, [z1 , z2 , ...] # BALL(R (Z)), t [x1, x2, ...] # BALL(C (Y )). These types of decompositions appear all over the literature concerning the Haagerup tensor product. Define f( y)= t t [(x1, y), (x2 , y), ...] and g([a1, a2, ...] )=k=1 zkak. The ``finite trunca-

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K(Cn(Y), Cm(Z))$Cm(Z)hA Rn(Y )$(Cm h Z)hA (Y h Rn)

$Cm h (ZhA Y )h Rn$Cm h K(Y, Z)h Rn

$Mm, n(K(Y, Z)).

Using these observations, we see that K(Y) is an operator inductive limit in the sense of Section 2 of ``finite dimensional matrix algebras over A.'' To see this notice that the maps ,; , ; , , ; , ; may be tensored to give completely contractive factorizations \; : Mn(;)(A) Ä K(Y) and %; : K(Y) Ä

Mn(;)(A). Namely, set \;=; hA , ; : Cn(;)(A)hA Rn(;)(A) Ä YhA Y and %;=,; hA ; : YhA Y Ä Cn(;)(A)hA Rn(;)(A). Although the %; are not homomorphisms, it can be shown that they do satisfy the important ``asymptotic homomorphism'' condition in Section 2, so that K(Y) is the operator algebra inductive limit of Section 2 of the Mn(;)(A) algebras.

We can also show, as in [32], that YhA CBA(Y, A) is completely isometrically isomorphic to a subalgebra of CBA(Y, Y) via the obvious map. We omit the proof since we shall not need this fact here. Hence this algebra is also an operator algebra containing K(Y) which in general has only a one-sided c.a.i. and is not equal to K(Y).

4. Pre-rigged Modules and Direct Sums

It is important to be able to treat Y as purely an algebraic module and later to induce the operator space structure which makes it a rigged module. As an example we shall give a first approach to the direct sum of rigged modules (see also Section 7).

Definition 4.1. Suppose A is a (not necessarily complete) operator algebra and that Y0 is a right A-module and suppose that there is a net of positive integers n(;) and right A-module maps ,; : Y Ä Cn(;)(A) and ; : Cn(;)(A) Ä Y such that

(1$) the composition ,; # is completely contractive for each ;, #.

Also sup; &,;( y)&< for all y # Y0 .

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(2$) after endowing Y0 with the initial uniform preoperator space structure induced by the maps ,; , then ; ,; Ä IdY strongly on Y0 .

We may then give Y0 the inductive limit module structure as in Section 2, and complete it to obtain Y. Suppose, further, that Y satisfies items (3) and

(4) of Definition 3.1. Then we say that Y0 is a right pre-A-rigged module. Thus to make the transition from Definition 4.1 to Definition 3.1, we define matrix norms on Mn(Y0) as in Section 2 by

&[ yij]&=sup &[,;( yij)]&. ;

This is simply an inductive limit structure which we are putting on Y0 , and thus Y0 is an operator space as in Section 2. It is straightforward to check that with these matrix norms, the completion of Y0 is an operator

A-module Y, each ,; and ; is completely contractive, and that ,# ; ,; Ä ,# uniformly in the completely bounded norm. We observe that in the cases which we have encountered, it is unne- cessary to first complete and then check conditions (3) and (4). Condition

(3) can be ensured at the uncompleted stage by insisting that the maps ; are ``left multiplication by a row'' with entries in Y0 ; that is, they are essential in the appropriate uncompleted space of maps. Similar remarks apply to condition (4). We state the definition as such to ensure maximum generality. Consider a finitely generated projective (in the purely algebraic sense) module Y over an operator algebra A. From algebra we know that there is an integer n, and module maps ,: Y Ä Cn(A) and : Cn(A) Ä Y such that b ,=IdY. However, we do not know in general when it may be arranged so that , b is completely contractive, or it has cb norm <1+= for a given =>0. When this can be arranged then Y can be given a rigged structure. We now construct the c-direct sum Äc of right A-rigged modules. We first note that for standard right A-rigged modules there is no obstacle: we c c c c simply define H A Ä K A=(HÄK)A, where HÄK is the Hilbert space direct sum. Similarly, for 3, or any number, summands. For arbitrarily rigged modules Y and Z over A we must be very careful; the ``obvious'' direct sum (defined for instance for Y/B(H1) and Z/B(H2) by setting c YÄ Z to be the elements of B(H1 ÄH2) which may be realized as 2_2 matrices with 11 entry from Y, 21 entry from Z, and other entries zero) is not well defined (see the discussion of this in [12]).

Let [Yk]k # 4 be a family of right A-rigged modules. Then we may write k k k k for each k # 4: Yk= ([Y;k], [,;k], [;k]). Here Y ;k=Cn(;k)(A), and we may suppose that ;k is in the directed set 2k .

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Form the restricted direct product 2 of the directed sets 2k here. That is, we consider the set of functions ; with domain a finite subset of 4 and with range contained in the disjoint union of the directed sets, such that

;(k)#2k for all k. This is again a directed set with the natural ordering, with typical element ;=(; )m . Thus (; )m (; )m$ if and only if kr r=1 kr r=1 kr$ r=1 [k , ..., k ] [k$ , ..., k$ ] and ; ; whenever k =k$ . Write n(;)= 1 m 1 m$ kr ks$ r s m m r=1 n(;kr), and for each ;=(;kr)r=1 consider the standard rigged module direct sum C (A)=C (A)Äc }}}Äc C (A). n(;) n(;k1) n(;km) Consider the (restricted) algebraic direct sum Y0=k Yk , together m kr with the obvious maps ,;= , : Y0 ÄCn(;)(A), and ;= r=1 ;kr m kr : Cn(;)(A)ÄY0. It is necessary to verify that the maps ,; , ; r=1 ;kr satisfy the conditions of Definition 4.1. For instance, we verify that ,; b # is a complete contraction. For notational simplicity, assume ;=#. Observe k k t ;k ;k t that ,;k ;k([a1, ..., an(;k)] )=[(xi , yj )][a1, ..., an(;k)] . The square matrix here has entries in A and has norm 1. This is true for all the summands occurring in ,; , ; , and so

m t ;kr ;kr t ,; ;([a1, ..., an(;)] )= [(xi , yj )] [a1, ..., an(;)] , \r=1 +

m where r=1 is the diagonal sum of the m square matrices concerned. Since each of the square submatrices has norm 1, so has the full matrix, which establishes the desired result. c We write Y=k Yk for the completion of the restricted algebraic direct sum in the norm induced by the ,; maps, or in other words we give Y0 the inductive limit operator module structure as in Section 2. Then Y is a right A-rigged module. We call this the column direct sum, or c-direct sum (of right A-rigged modules). As in the remarks before Theorem 3.6 we obtain associated maps from standard left A-rigged modules into and from r k Y k , and we write X=k Y k for the inductive limit operator module structure induced by these maps. We also call this the r-direct sum (of left A-rigged modules). c t r It needs to be verified that (k Yk) $ k Y k completely isometrically. c Note that the canonical map 3: k Y k Ä (k Y k) is well defined because m if g=(gkr)r=1 # k Y k, then E*;k gk Ä gk uniformly, for each k=k1 , ..., km . m c t If we set f=r=1 gkr then E ;* f Ä f uniformly. Thus 3(g)#(k Yk) .If we recall that Y is normed by the maps ;, then we see that 3 is a com- r c t plete isometry of the subspace k Y k of k Y k into (k Yk) . On the c t other hand, (k Yk) is generated by elements in 3(k Y k), namely, by k c t$r the functionals f ;k . This establishes the isomorphism ( k Yk) k Y k. Write ik for the embedding of Yk into Y, and ?k for the projection of Y onto Yk . It is easy to see that both ik and ?k are adjointable and that @~ k is the r r projection of k Y k onto Y k, while ?~ k is the embedding of Y k into k Y k.

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The following observation is important.

Lemma 4.2. The c-direct sum does not depend on the particular asymptotic factorizations of the identity maps on the summands Yk .

Proof. This follows from a slight variation of our proof above that ,; b # is a complete contraction. Indeed, if we also have that Yk= ([Cn(:k)(A)], k k k k t [':k], [`:k]), then it is possible to write ' :k ;k([a1, ..., an(;k)] )= :k ;k t [(xi , yj )][a1, ..., an(;k)] , for all k, and the matrix here has entries in A and norm 1. Therefore as above, ': b ; is a complete contraction. By c Proposition 2.6 we see that the identity map from the direct sum Yk defined by the , and nets, to the direct sum defined by the ' and ` nets, is completely contractive. By symmetry they are completely isometric. K

This last result also follows from the universal property of c given in Section 7.

5. Equivalent Definitions of Rigged Modules

We state a sample of alternate definitions as theorems proving their equivalence to 3.1. Depending on the setting, different definitions will be most useful. For instance 5.3 is useful in Section 7, 5.1 was most convenient in [12], and 5.4 seems to be convenient when concocting finite dimensional examples. The last definition 5.10 relates a rigged module to its ``C*-module envelope.''

Theorem 5.1 [12]. Suppose that A and B are operator algebras (with c.a.i.'s), that Y is a B&A-operator bimodule, that X is an A&B-operator bimodule, and suppose that X and Y are essential with respect to the actions of both algebras. Suppose that there exist completely contractive pairings (}, }):X_YÄA and [}, }]:Y_XÄB which are ``balanced'' bimodule maps, and the second of which induces a quotient map on the Haagerup tensor product Yh X Ä B. We suppose also the purely algebraic relations (x, y)x$=x[y, x$] and y(x, y$)=[y, x] y$(for x, x$#X, y, y$#Y). Then Y is a right A-rigged module, and X is a left A-rigged module. Moreover K(Y)$B completely isometrically isomorphically and Y $XasA&B operator bimodules (completely isometrically). Conversely, every rigged module arises in this way.

The tuple (A, B, X, Y, ( }, } ), [ }, } ]) above is called a (P)-context in [12].

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Theorem 5.2. Suppose that we have the same hypothesis as the previous theorem, with the following relaxations. We merely require the module actions on Y and X, and the pairings (},}), [},}], to be jointly completely contractive in the sense of [14], as opposed to being completely contractive in the sense of Section 2 or [22]. We replace the hypothesis of the quotient map Yh X Ä B by requiring that some approximate identity [e;] for B has n each term expressible as e;=k=1 [yk, xk], where n and xk # X, yk # Y t depend on ; and where &[ y1 , ..., yn]&, &[x1 , ..., xn] &1. Then the same conclusions hold, and in fact the previous hypotheses of 5.1 hold automatically.

Theorem 5.3. Suppose that A is an operator algebra with c.a.i., Yisan essential right A-operator module, and X is an essential left A-operator module. Suppose that there is a pairing (},}):X_YÄA which is a completely contractive A-bimodule map. Then the module Haagerup tensor product

B=YhAX is a Banach algebra with multiplication ( yx)( y$x$)= y(x, y$)x$. Also B has a natural left action on Y and a right action on X, using the pairing (},}). Set Z to be the essential subspace of Y for this action, and set W to be the essential subspace of X. If B has a c.a.i. then Z is a right A-rigged module and W is a left A-rigged module. Moreover, Z $W as A-operator modules (completely isometrically), and K(Z)$B completely isometrically isomorphically. Conversely, every rigged module arises in this way. Remark. If the span of the range of ( }, } ) is dense in A then one can show that Z=Y and W=X. We omit the proof.

Theorem 5.4. Suppose that A and B are operator algebras (with c.a.i.'s), and that Y and X are operator spaces and B&A- and A&B- bimodules respectively, essential with respect to all actions. Suppose that there exist purely algebraic pairings (},}):X_YÄA and [},}]:Y_XÄB such that the space L of 2_2 matrices

a x \ y b+ with a # A, b # B, y # Y, x # X , is an algebra (as discussed in Section 2). Suppose further that L has a faithful nondegenerate representation on a Hilbert space which is completely isometric on each of the four corners. Sup- pose that each element in some c.a.i. [e;] for B may be written in L in the n n n form e;=k=1 ykxk, where &k=1 yk yk*&1 and &k=1 xk*xk&1. Here n and xk # X, yk # Y depend on ;, and the adjoints V are taken with respect to the Hilbert space L lives on. Then Y is a right A-rigged module and X

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Theorem 5.5. Suppose that L is an operator algebra with c.a.i. and suppose that p and q are two orthogonal projections in the multiplier algebra

M(L) of L such that p+q=1M(L). Let A=pLp, B=qLq, Y=qLp, X=pLq. Suppose that each element in some c.a.i. [e;] for B may be written n as e;=k=1 ykxk for xk # X, yk # Y, where n, xk , yk depend on ;, and n n &k=1 yk yk*&1 and &k=1 xk*xk&1. Then Y is a right A-rigged module and X is a left A-rigged module. Moreover, Y $X, and K(Y)$B as in the theorems above. Conversely, every rigged module arises in this way. Before we discuss the proofs, let us demonstrate how to verify the conditions of 5.4 in concrete cases.

Example 5.6. Let A be the algebra of n_n upper triangular matrices, let Y=B=Mn , and let X be the space of n_n matrices supported on the first row. Then the set of matrices

a x \ y b+ with a # A, b # B, y # Y, x # X, is a concrete subalgebra of M2n , and the ideal in B spanned by terms yx is clearly equal to B. The identity In of B, n equals k=1 Ei1E1i, which shows that the conditions of 5.4 are met. We will merely sketch the proofs of these theorems. Viewing the theorems as definitions it is clear that 5.4 implies 5.5.

Conversely, 5.5 implies 5.4 after we note that if [e:] is a c.a.i. for L then

[pe: p] and [qe:q] are c.a.i.'s for A and B, respectively, and, moreover, these latter c.a.i.'s converge to the identity map when multiplied by some- thing in Y or X appropriately, so that X and Y are essential bimodules. It is clear from Lemma 2.9 in [12] that 5.4 implies 5.1. That 5.3 implies 5.1 is fairly clear, since B in 5.3 is an operator algebra because its multiplication is completely contractive (by the associativity of the Haagerup tensor product), and the rest of the conditions in 5.1 are obvious with [ y, x] defined to be yx. Conversely, assuming 5.1 it follows from [12, Theorem 3.5] that the Banach algebra in 5.3 has a c.a.i., whence 5.3. Clearly 5.1 implies 5.2, since every completely contractive bilinear map is jointly completely contractive, and the approximate identity condition is Lemma 2.9 in [12] again. The converse is shown in [10] (the trick is essentially that of [12, Lemma 4.7]).

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We leave it as an exercise for the reader that 5.1 implies 3.1 (or see [12]). The reverse, that 3.1 implies 5.1 or 5.3 can be seen immediately from Theorem 3.10, if not from more elementary considerations. That is, if Y is an A-rigged module then (A, K(Y), Y , Y) is a ``(P)-context.'' The chain will be complete if we can show that 3.1 implies 5.4 or 5.5. The following argument yields an important realization of the linking operator algebra:

Theorem 5.7. Let Y be a rigged module over an operator algebra A with c.a.i. Then K(AÄc Y) is the (unique) linking operator algebra for the (P)-context (A, K(Y), Y , Y). That is, L=K(AÄc Y) satisfies the conditions of 5.5 with pLp$A, qLq$K(Y), qLp$Y, and pLq$Y completely isometrically isomorphically, for the natural complementary projections p, q.

Proof. We write iA , iY (resp. ?A , ?Y) for the natural inclusions (resp. c c projections) of A and Y into AÄ Y (resp. from AÄ Y). Set p=iA b ?A , q=iY b ?Y . Note that A$K(A). There is a completely contractive c homomorphism _:K(A) Ä K(AÄ Y):T[iAT?A . However, there is also c an obvious retraction K(AÄ Y) onto K(A) given by T [ ?A TiA , which is also completely contractive. Hence _ is a complete isometry. By the same argument K(Y) embeds into K(AÄc Y), via the c homomorphism %: T [ iY T?Y ; and Y$K(A, Y) embeds into K(AÄ Y), via the module map : T [ iY T?A and similarly for X$K(Y, A). The span of the images of the four embeddings is clearly all of K(AÄc Y), and the induced map a x a#A, b#B, x#X, y#Y ÄK(AÄc Y) {\y b+} = is an algebra isomorphism which is completely isometric on each of the four corners. Hence, K(AÄc Y) is a concrete representation of the linking algebra as an operator algebra. The approximate identity condition in 5.4 or 5.5 is clearly satisfied. By the argument in [12, Section 5] this representation is unique. K It follows from the uniqueness above, for example, that if Y is a Hilbert C*-module over a C*-algebra, then K(Y) is the usual ``imprimitivity'' C*-algebra. In fact more is true.

Theorem 5.8. The class of rigged modules over a C*-algebra coincides with the class of Hilbert C*-modules (with their canonical operator space structure). In particular, if Y is a rigged module over a C*-algebra A, then Y is a genuine Hilbert C*-module over A, K(Y) is a C*-algebra, and the inner product on Y is given by ( y1 | y2) =y1*y2 , where the adjoint and multiplication is performed in the linking operator algebra.

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Proof. This is essentially in [12] but for completeness we give the argument. Form the linking operator algebra L as in Theorem 5.7, say, and suppose that it is represented nondegenerately on a Hilbert space H, say. This allows us to write the pairings and module actions as concrete ; ; operator multiplication in B(H). If e;=i yi xi is a c.a.i. for B=K(Y) ; ; as in Theorem 5.4, then e;*e;i xi *xi 1. Thus for b # B, we have ; ; ; ; b*(1&i xi *xi )bb*(1&e*; e;)b Ä 0. Hence - 1&i xi *xi bÄ0, which ; ; implies that (1&i xi *xi )b Ä 0. Since B acts nondegenerately on Y we ; ; ; ; have that i xi *xi y Ä y for all y # Y. Thus i (xi y)* x i Ä y*. Since ; (xi y)* # A*=A, it follows that Y*/X, where X=Y . Similarly, X*/Y. Thus Y=X*, and so B=(YX)& is self-adjoint, and Y is a Hilbert

C*-module with A-valued inner product (y1 |y2)=y1*y2. Notice that the operator space structure induced by the inner product coincides with the original operator space structure on Y. K

Corollary 5.9. If A is an operator algebra with c.a.i. and if C is a C*-algebra containing A, which is generated as a C*-algebra by A, then C is an essential A-operator bimodule. That is, C=CA=AC=ACA. Conse- quently we also have C=A*C=CA*. Proof. If we set A$=C, Y=(AC)&, X=(CA)&, B=(ACA)&, then it is easy to check that the conditions of the equivalent characterizations of rigged modules (e.g., Theorem 5.5) are satisfied. Hence Y is a C*-module over C, and consequently B is a C*-algebra. Since B is generated by A we get B=C. Since ACA/AC/C we get Y=C. Thus A acts essentially on the left of C, so by Cohen's factorization theorem AC=C, and similarly CA=C.It follows that every c.a.i. for A is also a c.a.i. for C. K In a discussion with V. Paulsen, we noticed that if one replaces C in the proof above with any C*-algebra B containing A, the proof shows that the closure of ABA is a C*-algebra, and the closure of AB is a right C*-module over B. In fact (AB)&=(CB)& (since (AB)&/(CB)&=(ACB)&/ (AB)&) and also (ABA)&=(CBC)&.

Theorem 5.10. Suppose that A is an operator algebra with c.a.i., that B is a C*-algebra generated by the unitization A+ of A, and that Z is a C*-module over B. Suppose that Y is a closed A-submodule of Z. Let W= [z # Z: (z| y) # A \y # Y]. Suppose that there exists a c.a.i. for K(Z) con- n sisting of elements of form k=1 |yk)(wk| in D=Span[| y)(w|:w#W, n n y#Y], with k=1 |yk)( yk|1 and k=1 |wk)(wk|1. Let C be the closure of D in K(Z). Then Y is a right A-rigged module and W is a left A-rigged module. Moreover, Y $[wÄ # Z : w # W] as A-operator modules (completely isometrically) and K(Y)$C completely isometrically isomorphically. Conversely, every rigged module arises in this way.

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Proof. To prove the forward implication we verify that the conditions of Theorem 5.1 are satisfied. Consider the linking C*-algebra L of Z. Note that W is a right module over A*, adjoint taken in B. Consider X=W*, where the adjoint is taken in L,soX/Z*=Z . Then X is a left A-module under the action aw*=(wa*)*. The natural pairing X_Y Ä A is completely contractive, since it is simply multiplication in L. The space C is clearly an operator algebra, being a closed subalgebra of K(Z). Moreover, C contains a c.a.i. for K(Z). Thus C acts essentially on the left of Y and on the right of X. The pairing Y_X Ä C:(y, x)[|y)(x*| is also completely contractive, being multiplication in L. Moreover, this last pairing is also a quotient map by Lemma 2.9 in [12]. We still need to verify that A acts essentially on X. However for c=|y)(w|#C and x # X we have xc=(xy)w*. Since xy # A we have e* xc Ä xc for a c.a.i. [e*] for A. Thus the same thing holds for any c # C, so that e* x Ä x for any x # X since C acts nondegenerately on X. Thus A acts essentially on X. We obtain Y $X=W* from Theorem 5.1. The converse assertion shall be proved in Section 6 after Theorem 6.8. K

We shall see many corollaries of this in the next two sections.

6. Induced Representations and Tensor Products of Rigged Modules

We first point out another connection between what we have just done and Morita equivalence of operator algebras.

Theorem 6.1. If A and B are strongly Morita equivalent operator algebras in the sense of [12] (see also Section 8), then their categories of right rigged modules are isomorphic, under an isomorphism which preserves the algebraically finitely generated projective rigged modules. Moreover this isomorphism is implemented by ``tensoring with the equivalence bimodule'' and similarly for the categories of left rigged modules.

Proof. Suppose that (A, B, X, Y) is a Morita context. If NA is a right

A-rigged module then as in [12], NhA X is an essential right-B-module. Moreover, it follows from 5.3, say, that NhA X is a right B-rigged module with K(NhA X)$K(N), since (NhA X)hB (YhA N )$ NhA (XhB Y)hA N $NhA AhA N $NhA N $K(N). From this equality we see that K(N) is unital if and only if K(NhA X) is unital. There is an obvious ``inverse functor'', as in [12, Section 3], so that the categories are isomorphic. K

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It follows from the ``cb-version'' of this argument that strong Morita equivalence of unital operator algebras preserves K0 groups (although this may also be deduced from a stable isomorphism theorem). This uses the fact that tensor products distribute over direct sums, which we shall see in Section 7. In the nonunital case one may have to pursue an argument such as may be found in [31]. We introduce a little more notation. The following result is well known in the C*-case.

Theorem 6.2. Let A and B be operator algebras, and let % be a completely contractive homomorphism %: A Ä M(B). The following conditions are equivalent:

(1) [%(e:)] converges strictly to the identity in M(B), for any bounded approximate identity [e:]ofA,

(2) [%(u:)] converges strictly to the identity in M(B), for some norm bounded net [u:]inA, (3) any element of B may be expressed as a product %(a)b and also as a product b$%(a$), for some a, a$#A and b, b$#B. (4) % has a unique completely contractive unital extension homomorphism %1 : M(A) Ä M(B), such that when restricted to bounded subsets, %1 is continuous with respect to the strict topologies. Proof. Clearly (1) implies (2), and (3) and (4) each imply (1). Define a module action of A on B by a } b=%(a)b. Then conditions (1) or (2) imply that the action is essential, and hence half of (3) follows from Cohen's factorization theorem. The other half is identical. To obtain (4) we may follow the algebraic argument of [57]. Namely define %1 , the extension of %, as mapping into the left multipliers of B by the formula %1(c)(%(a)b)=%(ca)b. Note that any extension satisfies this relation. This observation gives, using (3), the uniqueness of extension. To prove that the extension is well defined suppose that %(a)b=%(a$)b$. Then for c # M(A), d # A, e # B we have e%(d) %(ca)b=e%(dc) %(a)b= e%(dc) %(a$)b$=e%(d) %(ca$)b$. Since B consists of just such terms e%(d)by

(3), we see that %(ca)b=%(ca$)b$, which shows that %1(c) is well defined as a left multiplier. Indeed its clear from the first equality in the line above that %1 maps into the double centralizers, that is, into the multiplier algebra. The map %1 is completely contractive: if [cij]#Mn(M(A)), [bkl]#

Mn(B), and if &[cij]&, &[bkl]&<1, then by Cohen's theorem there exists a # A,[b$kl]#Mn(B), with &a&<1, &[b$kl]&<1 and [bkl]=[%(a) b$kl].

Hence &[%1(cij)(bkl)]&=&[%(cij a) b$kl]&<1 which establishes that %1 is completely contractive. The statement in (4) about the continuity also uses a trivial application of Cohen's theorem. K

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Note also that the proof works for completely bounded homomorphisms, and the unital extension has no increase in the cb-norm. We call a morphism satisfying the equivalent conditions above an essential homomorphism. In the C*-theory they have also been called (S)-morphisms [34] or nondegenerate morphisms. They appear in many places in the literature these days. Their importance seems to be that in areas of noncommutative topology the usual morphisms A Ä B do not suffice. For instance, it is well known that the category of locally compact spaces and continuous maps is (anti-)equivalent to the category of commutative C*-algebras and essential homomorphisms. Since %: A Ä M(B) has an extension M(A) Ä M(B), if ?: D Ä M(A)is another essential homorphism, then we can ``compose'' to get an essential homorphism which we write as % b ?: D Ä M(B). It is easy to see that `òperator algebras with c.a.i.'s'' and `èssential homomorphisms'' form a category, which we write as OA. If Z is a right B-rigged module and if %: A Ä B(Z) is a completely contractive homomorphism, then we can define a completely contractive action of A on Z by a } z=%(a)z. The formula w } a=w b %(a) defines a completely contractive right action of A on Y . We omit to write the `ò'' in what follows.

Proposition 6.3. If A and B are operator algebras, if Z is a right B-rigged module, and if %: A Ä B(Z) is a completely contractive homomorphism then % is essential if and only if %(e:)z Ä z, and w%(e:) Ä w for all z # Z, w # Z , that is, if and only if % induces an essential module action on both Z and Z . This follows in the one direction from the fact that K(Z) acts essentially on both Z and Z , and Cohen's theorem, and in the other direction from the fact that K(Z) is the closure of the finite rank operators. We proceed to discuss ``change of rings,'' which is related to the topic of ``induced representations.'' As above, if %: A Ä M(B) is an essential homomorphism we may define a module action of A on B by a } b=%(a)b. Then B is an essential left

A-operator module, and consequently AhA B$B completely isometrically from [12, Lemma 2.5]. From this it follows that

Cn(A)hA B$Cn h AhA B$Cn(B) completely isometrically.

Suppose that Y is a right A-rigged module, with factorization maps ,; and ; . Suppose that [e*] is a c.a.i. for B, and write L* for the map

B Ä B: b [ e* b. Let ,$;, * : YhA B Ä Cn(;)(B) be defined by following

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,; IdB with L* (after identifying Cn(A)hA B with Cn(B)). Similarly define $;, * : Cn(B) Ä YhA B by following L* with ; IdB . It is easy to see that conditions (1) and (3) of Definition 3.1 are met. It is also not too difficult to see that (2) and (4) are met if we use the iterated limit lim; lim* . However this is sufficient for our theory in Sections 35 to work. In any case it is clear that C=(YhA B)hB (BhA Y )=YhA BhA Y is a Banach algebra; the natural multiplication on this algebra is completely contractive, since the natural pairing of (BhA Y )_(YhA B) into B is completely contractive as may be seen by following the factorization:

(BhA Y )h (YhA B) Ä BhA (Y h Y)hA B Ä BhA AhA B Ä B. The new feature is that the contractive ``approximate identity'' is indexed by (;, *), and we need to apply the limits in the order lim; lim* .By Lemma 2.1 we can reindex to obtain a genuine approximate identity indexed by a single directed set. It is also easy to see that this algebra C acts nondegenerately on both

YhA B and BhA Y . We now can appeal to Theorem 5.3 to conclude that YhA B is a genuine rigged module over B. We have proved the following. Theorem 6.4. If A and B are operator algebras, and if %: A Ä M(B) is an essential homomorphism, and if Y is a right A-rigged module, then YhA B is a right B-rigged module. Also we have that t (YhA B) $BhA Y and K(YhA B)$YhA BhA Y (completely isometrically isomorphically). B B We write Y or Y% B for YhA B. We also write X or B% Y for BhA Y . The construction Y [ Y B is called ``change of rings,'' and is, by elementary functorial properties of the Haagerup tensor product, functorial (see Theorem 6.6 below). An important case is where A is a subalgebra of B. More generally, suppose that Y is a right A-rigged module, that Z is a right B-rigged module, and that %: A Ä B(Z) is an esential homomorphism. We may then form Y K(Z) and change rings to K(Z). However, we may obtain an induced representation of B as follows. As described above, Z becomes an essential left A-operator module under the action a } z=%(a)z. We form the Haagerup module tensor product YhA Z, which we also write as Y% Z. We claim that this is a right B-rigged module. To see this, form Z hA Y , which we also call Z % Y , and define a B-valued pairing

(wx, yz)B=(w, %((x, y)A)z)B of Z hA Y with YhA Z. This is completely contractive, as may be seen by following the factorization: (Z % Y )h (Y% Z) Ä

Z hA (Y h Y)hA Z Ä Z hA AhA Z Ä Z hA Z Ä A.

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Next observe that (Y% Z)hB (Z % Y )$YhA (ZhB Z )hA Y $

YhA K(Z)hA Y . This latter object is an operator algebra by Theorem 6.4, which is easily seen to act nondegenerately on Y% Z and on Z % Y , so that by Theorem 5.3 it follows that Y% Z is a right B-rigged module and, moreover, it follows that

t (Y% Z) $Z % Y and that

K(Y% Z)$YhA K(Z)hA Y completely isometrically isomorphically. One may also realize this by explicitly constructing factorization maps as in the proof of Theorem 6.4. Notice also that if Y and Z are algebraically finitely generated and projective, then so is Y% Z.

We call Y% Z the (interior) tensor product of rigged modules. Note that we have retained the original ring acting on Z, but changed the endomorphism ring.

Theorem 6.5. The (interior) rigged tensor product described above is associative. That is, if Y is a right A-rigged module, if Z is a right B-rigged module, if V is a right C-rigged module, and if %: A Ä B(Z) and \: B Ä B(V) are essential homomorphisms, then (Y% Z)\ V$Y% (Z\ V) completely isometrically isomorphically.

This follows from the associativity of the Haagerup tensor product. By the functorial property of the Haagerup tensor product, the (interior) tensor product of rigged modules is functorial.

Theorem 6.6. Suppose that Y1 and Y2 are right A-rigged modules, that Z1 and Z2 are right B-rigged modules, and that %: A Ä M(B) is an essential homomorphism. If f: Y1 Ä Y2 is a completely bounded and adjointable right A-module map and if g: Z1 Ä Z2 is a completely bounded and adjointable right B-module map, which is also a left A-module map, then fg : Y1 % Z1 Ä Y2 % Z2 is a completely bounded adjointable right t B-module map, with & fg&cb& f &cb&g&cb . Further,(fg) =g~ f . Proof. Note that g~ is a right A-module map. We can certainly define fhA g: Y1 hA Z1 Ä Y2 hA Z2 , and g~ hA f : Z 2 hA Y 2 Ä Z 1 hA Y 1 and these will be completely bounded by Theorem 2.3 in [12], with the requisite cb norm bounds. We need to check that fhA g is adjointable.

For y # Y1 , z # Z1 , w # Z 2 , x # Y 2 we have

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(wx, f( y)g(z))B=(w, %((x, f(y))A) g(z))B=(w%(( f(x), y)A), g(z))B

=(g~ ( w%(( f (x), y)A), z)B=(g~ ( w ), %(( f (x), y)Az)

=(g~ ( w ) f ( x ), yz)B . K As an example of the interior tensor product, if A=C,ifYis a right A-rigged module (therefore Y=H c for some Hilbert space H), and if Z is c a right B-rigged module over an operator algebra B, then Y% Z=H Z quite obviously (recall that H c =H c Z=H c Z). In contrast to this Z h we have the following

Theorem 6.7. If A is an operator algebra, if Y is a right A-rigged module, if B=C, and if H is the Hilbert space of a nondegenerate represen- c tation % of A, then Y% H is a Hilbert space. More generally, if B is a - - C* algebra and if Z is a right Hilbert C* module over B, then Y% Z is a right Hilbert C*-module over B ( for any essential homomorphism %: A Ä B(Z)). This seems to give a method to manufacture Hilbert C*-modules. The following is an important special case. Suppose that Y is a rigged module over an operator algebra A with c.a.i., and suppose that B is a unital C*-algebra containing the unitization A+ of A and which is generated as a C*-algebra by A. Let C be the C*-subalgebra of B generated by A.By Corollary 5.9, C is an essential A-operator bimodule, so that AC=C=AB, the last equality since AB/C=AC/AB. We consider Y as a rigged module over A+, and B as an essential left A+ module and form the inte- + rior tensor product Z=Y% B (where % is the inclusion of A in B). This coincides with YhA C, since

YhA+ B$(YhA A)hA+ B$YhA (AhA+ B)

& $YhA (AB) =YhA C.

However, in what follows we usually work with Z=Y% B, since B has identity.

Theorem 6.8. With the notation above Z=Y% B is a genuine Hilbert C*-module over B. Moreover, Y (resp. Y ) is contained in Z (resp. the conjugate Hilbert C*-module Z ) completely isometrically (and as A-modules), and K(Y) is completely isometrically isomorphic to a subalgebra of the C*-algebra K(Z), and this subalgebra contains a c.a.i. for K(Z). Proof. We may assume w.l.o.g. that A is unital. Certainly Z is a Hilbert C*-module over B. Consider the map i: Y Ä Z : y [ y1. This is a completely contractive right A-module map. However the discussion preceding

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Theorem 6.4 informs us that Z as a rigged module is determined by the factorization maps ,$;=,; IdB , and $;=; IdB . However, note that

,$; b i maps into Cn(;)(A) and equals ,; . Since the maps ,$; determine the operator space structure, this forces i to be completely isometric. A similar argument applied to the map j: Y Ä Z : x [ 1x shows that j is a completely isometric left A-module map. A similar argument (using the factorizations of K(Y) discussed at the end of Section (3) applied to the map i j: K(Y) Ä K(Z) shows that it too is completely isometric. K

As we said in the introduction, this allows us to define an actual (B valued) inner product on Y.

Example 6.9 Let f be an invertible function in H (D) which is not in

A(D), such that there exist functions hn in A(D) with |hn f | Ä 1 and &1 &1 |hn f | Ä 1 uniformly (see Example 8.2 in [12]). Then Y=A(D) f is a rigged A(D)-module (actually an equivalence bimodule for a Morita equivalence of A(D) with itself ). However, Y is not free (that is, it is not completely isometrically isomorphic to a standard A(D)-rigged module), nor is it even completely isometrically isomorphic to a direct summand of a free. Here Y =A(D) f &1. In the ``cb-version'' of the theory this example would get lost, since this module is cb-bicontinuously isomorphic to A(D). Notice that if we let % be the inclusion of A(D)inC(D ), and change C(D) rings to obtain a right C(D )-rigged module V=Y =Y%C(D ), then V is a Hilbert C*-module over C(D ) (that is, it has an inner product which determines an operator space structure which coincides with its rigged module operator space structure). One can show that V=C(D ), so that we have not produced a new example of a Hilbert C*-module.

We can now complete the proof of Theorem 5.10. If Y is an A-rigged + module and B is a unital C*-algebra generated by A then let Z=Y% B as above. Then Y /Z and K(Y)/K(Z), and some c.a.i. for K(Y) is also one for K(Z). It is easy to see from the uniqueness of the linking algebra [12] that the linking algebra for Y may be viewed as the obvious subalgebra of the linking algebra L for Z. This was noted first in [Na]. We now show that Y regarded as a subset of L, coincides with [z*#Z*: (z| y)=z*y # A \y # Y], adjoints taken in L. Certainly the first is a subset of the second since Y /Z . However, if z*y # A for all y # Y, then z*(yx)#Y for all y # Y, x # Y ,soz*E;#Y, where E; is the common approximate identity. Thus z*#Y . The rest of 5.10 is straightforward. Theorem 5.10 has the potential to be the most effective tool for dealing with rigged modules. The proof actually shows that if w # Z and if (w| y1) #A for all y # Y, then there is an x # Y with (x, y)= (w| y1) \y#Y.

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Proposition 6.10. If Y is a rigged module over A, if A is represented c nondegenerately on a Hilbert space H, if 8: Y Ä B(H, YhA H ) is the map c 8( y)(`)=y`, and if 9: Y Ä B(YhA H , H) is the map 9(x)(y`)= (x, y)`, then 8 and 9 are completely isometric A-module maps. Also if c 3: B(Y) Ä B(YhA H ) is 3(T)(y`)=T(y)`, then 3 is a completely isometric homomorphism. Moreover, the linking algebra for Y is completely c isometrically isomorphic to a subalgebra of B(HÄ(YhA H )) via the maps above. Proof. This was noted as ``Method 1'' at the end of Section 5 in [12], without a proof. To deduce the result quickly from what we have done here, observe that if B is the unital C*-algebra in B(H) generated by A+, c c c then (YhA+ B)hB H $YhA+ H $YhA H . From this and Theorem 6.8, it follows that it is sufficient to prove the proposition for C*-modules, replacing the Haagerup tensor product with the interior tensor product [11]. But this is well known. K

Proposition 6.11. Suppose that T: Y Ä Z is a completely bounded module map between rigged modules over an operator algebra A. Suppose + that B is a unital C*-algebra generated by A . If TId: Y% B Ä Z% B is adjointable, then so is T.

Proof. For w # Z , y # Y we have for some zw # Z% B that (w, Ty)= (zw | Ty1B) =((TIdB)* (zw)|y1B). This is always in A, so by the comment after the completion of the proof of Theorem 5.10 above there is an x # Y with ((TIdB)* (zw)|y1B)=(x, y) for all y # Y. Thus T is adjointable. K

Theorem 6.12. Suppose that Y is a rigged module over an operator algebra A, implementing a Morita equivalence in the sense of [12] between A and B$K(Y). Suppose that A/B(H) nondegenerately and that C*(A) (resp. C) is the C*-algebra generated by A (resp. A+) in B(H). Let %: A+/C be the inclusion. Then C*(A) is strongly Morita equivalent to the - - C* algebra K(Y% C)$Y% C% Y . This latter C* algebra is generated as a C*-algebra by K(Y). - Proof. Let Z=Y% C,aC*module over C. The canonical pairing

Z _Z$(C% Y )_(Y% C) Ä C obviously has range in C*(A), since the two middle terms combine to give an element of A. Thus the closure of the range of this pairing is a two-sided ideal in C*(A). However this ideal contains A, as may be seen by considering the restriction of the pairing to terms of form (1x)_( y1). Thus the ideal is all of C*(A). Thus C*(A) is Morita equivalent to K(Z). The latter contains K(Y). The discussion below shows that K(Y) generates K(Z). K

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We can put the above into the following setup. Suppose that L is the linking algebra for a Morita equivalence of operator algebras A and B with c.a.i., which has been represented in B(HÄK) as 2_2 matrices

A X \Y B+ in the standard way so that (AH)&=H, and (BK)&=K. Let A$ (resp. B") be the C*-algebra generated by A in B(H) (resp. by B in B(K)). Set Y$=(YA$)&, X$=(A$X)&, B$=(YA$X)& and set L$ to be the subalgebra

A$ X$ \Y$ B$+ of B(HÄK). It is easily checked that this satisfies Theorem 5.5, say, so that it follows from 5.8 that Y$isaC*-module over A$ and, also, that L$isa C*-algebra. Hence L$ is the C*-subalgebra of B(HÄK) generated by L. On the other hand, by symmetry, setting Y"=(B"Y)&, X"=(XB")&, A"=(XB"Y)&, we have that

A" X" L"= \Y" B"+ is also the C*-subalgebra of B(HÄK) generated by L, so it equals L$. From this we read off that A$=A", B$=B", X$=X", Y$=Y". Conse- quently, A$ and B" are strongly Morita equivalent as C*-algebras, via the equivalence bimodule Y$, and K(Y$) is generated by K(Y)asaC*-algebra.

For a general rigged module over A, K(Y% C*(A)) is not generated as aC*-algebra by K(Y) (consider rigged modules of the type discussed in Example 3.3 where K(Y)=C (see also [60])). Here is another quick corollary: a rigged module Y is countably generated iff K(Y) has a countable c.a.i.. For if Y is countably generated - then so is the C* module Z=Y% B, where B is a unital C*-algebra generated by A+. Thus K(Z) has a countable c.a.i.. Then by an elementary argument we can find a subsequence of the common c.a.i. of K(Y) and K(Z) which is still an approximate identity for K(Z). The converse is easier; if K(Y) has a countable c.a.i. then we can find a countable c.a.i. in the dense subalgebra YY , showing that Y is countably generated. As in the C*-case we can also define an exterior, or spatial, tensor product of rigged modules. Namely if Y is a right A-rigged module, and if Z is a right B-rigged module, then the algebraic tensor product YZ is a right (A B)-module. Note that C (A) C (B)$C (A B) com- n m nm pletely isometrically. If ,; , ; are factorization nets for Y, and if `: , ': are

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factorization nets for Z, then we can define factorization maps ,; `: and ' of YZ through spaces C (A) C (B), and check (as we ; : n(;) m(:) did in Section 4 for direct sums), that the conditions of Definition 4.1 are met. We write Y Z for the (A B)-rigged module which results, and call it the exterior tensor product of rigged modules. It is seen easily from the functorial property of the spatial operator space tensor product (see [44, 14]) that as an operator space the exterior tensor product of two rigged modules is the spatial operator space tensor product. That is, one could define the exterior tensor product to be the obvious submodule of the spatial operator algebra tensor product of the linking operator algebras. This is not pointed out in the C*-literature.

7. Direct Sums, Complementability, and K-Groups

Lemma 7.1. (1) Let P: Y Ä Y be a completely contractive idempotent module map on a rigged module Y. Then the range of P is a rigged module of Y. Also P is adjointable both as a map into Y and into P(Y). The dual module (P(Y))t of P(Y) can be identified completely isometrically with the submodule P (Y ) of Y , with the dual pairing being the restriction of the pairing Y _Y Ä A. (2) Suppose that Z is a rigged module over A, that Y is an operator module over A, and that i: Y Ä Z and ?: Z Ä Y are completely contractive

A-module maps with ? b i=IdY . Then Y is a rigged module over A, and i and ? are adjointable. Proof. Let B be a C*-algebra generated by A+. Tensoring with B in the Haagerup tensor product we obtain a contractive idempotent module map - PIdB on the C* module Z=YhA+ B. This is adjointable (see [11], for example), and in fact self-adjoint. Thus by Proposition 6.11 we see that P is adjointable as a map into Y. Set W=Ran P =[fbP:f#Y], this is a closed submodule of Y . It is an easy exercise to check that P(Y), W,(},}) satisfies the conditions of Theorem 5.3. This proves (1). For (2), note that i is a complete isometry onto a submodule Ran i of Y satisfying (1) with

P=i b ?. Thus Ran i and Y are rigged modules. Now iIdB , ?IdB are adjointable by Theorem 3.7 in [11], with (iIdB)*=?IdB . Thus i and ? are adjointable by Proposition 6.11. K

Let [Yk]k # I be a family of rigged modules over an operator algebra A. c We give a development of the direct sum k Yk which follows an approach taken in [42], which seems to be quicker and to give more information than the presentation in an earlier version of our paper. By - Theorem 5.10 the Yk may be regarded as A-submodules of C* modules - + Zk=Yk % B over some fixed C* algebra B generated by A . Here % is the

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+ c inclusion of A in B. We define Y=k Yk to be the obvious closed - c A-submodule of the C* module direct sum Z=k Zk . This is an essential r operator module over A. Let k Y k be the obvious closed left A-sub- - r r module of the (left) C* module direct sum W=k Z k= k (B% Y k). Let us write X for this submodule, an essential left A-operator module. The natural completely contractive pairing W_Z Ä B restricts to a pairing

(}, }):X_YÄA, which explicitly is ((xk)k ,(yk)k)=k (xk, yk). The closure of the span of terms |y)(x| for y # Y, x # X is thus a subalgebra C of K(Z). The map Y_X Ä C:(y,x)[|y)(x| is completely contractive, being restriction of the pairing Z_W Ä K(Z). Moreover, an obvious but tedious calculation shows that C has a c.a.i. (assembled from the c.a.i.'s for the K(Yk)) satisfying Theorem 5.1, and C acts nondegenerately on Y and X. Thus Y is an A-rigged module, and Y $X completely isometrically. It may seem as if the direct sum Y depends on the B chosen, but the following universal property shows that this is not the case.

Theorem 7.2. Let [Yk]k # I be a family of operator modules over an operator algebra A. Suppose Y is a rigged module over A, and that ik : Yk Ä Y and ?k: Y Ä Yk are completely contractive A-module maps such that ?k ij=$k, j IdYk for all k, j # I, where $k, j is the Kronecker delta. Then Yk is a rigged module, and ik , ?k are adjointable for all k. If k ik ?k converges point norm (that is, strongly) on Y, then there is a submodule N of Y which is also a rigged module so that Y is completely isometrically isomorphic to c c NÄ (: Y:). If, further, k ik ?k=IdY then N=(0).

Proof. By Lemma 7.1, each Yk is a rigged module, and ik , ?k are adjointable. Also Qk=ik?k is a contractive projection in B(Y) and, hence, is self-adjoint. In fact the Qk are mutually orthogonal projections, so any finite sum of these are contractive. Setting R( y)=k Qk(y) for y # Y we see easily that R is an idempotent, contractive, module map on Y. Lemma 7.1 shows that R and consequently also Q=1&R are projections in B(Y), and N=Ran Q is a rigged module over A. Let i: N Ä Y be the inclusion map and set ?=Q. The maps iIdB , ik IdB and ?IdB , ?kIdB satisfy the c c conditions of Theorem 3.7 in [11], so that (N% B)Ä (k (Yk % B))$ Y% B unitarily. This completely isometric isomorphism restricts to a com- c c pletely isometric isomorphism of NÄ (: Y:) (as defined above the c c theorem) onto Y. Explicitly this isomorphism T: NÄ (: Y:) Ä Y is T((n,(yk)k))=n+k ik(yk). K

Corollary 7.3. The c-direct sum is associative and symmetric. That c c c c is,(Y1ÄY2)ÄY3$Y1Ä(Y2ÄY3)completely isometrically, and c c Y1 Ä Y2$Y2 Ä Y1 completely isometrically. Similar relations hold for an arbitrary number of summands.

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The following is a functorial property of the direct sum.

Theorem 7.4. Suppose [Yk] and [Zk] are two families of right

A-rigged modules, and suppose that fk : Yk Ä Zk is a completely bounded adjointable map, for all k, and suppose supk & fk&cb< . Then c c f=k fk : k Yk Ä k Zk is a completely bounded adjointable map with & f &cb= supk & fk&cb . Moreover, if the number of indices k is finite, and if each fk is a compact adjointable map, then so is f. Proof. The first assertion follows from the analoguous (and - straightforward) result for C* modules be tensoring the maps with IdB as before. The adjointability of f is immediate. To prove the last assertion, it is sufficient by density to prove that the direct sums of two rank 1 operators is finite rank. But this is evident: | y)(x|Ä |z)(w|= |yÄ0)(xÄ0|+|0Äz)(0Äw|. K

If the Yk are closed submodules of a rigged module Y satisfying the property in Theorem 7.2 with N=(0), where ik are the inclusion maps then c we say that Y=k Yk internally. It follows that Yk is a rigged module for all k, and we say that each Yk is c-complemented in Y. Being c-complemented is equivalent to being the range of a completely contractive idempotent module map P, by 7.1. This P is in B(Y) and it is unique. For suppose that P and Q are completely contractive projections in B(Y), with PY=QY. Then in B(Y) we have PQ=Q and QP=P. However, B(Y) is an operator algebra, so that P, Q are selfadjoint projections, so P=Q. This shows that if Y is a c-complemented submodule of a right A-riggid module Z then there is a (necessarily unique) closed submodule N of Z such that Z= YÄc N (internally). Both Y and N are rigged modules. We call N the c-complement of Y in Z and write N=Z c Y. Let Y and Z be right A-rigged modules and suppose that i: Y Ä Z is a completely contractive module map. It is easy to see from 7.1 that the following are equivalent for a completely contractive module map j: Y Ä Z :

(1) i or j is adjointable and ( j(x), i( y))A=(x, y)A for all x # Y , y # Y. } (2) j is adjointable and ~ b i=IdY.

(3) i is adjointable and @~ b j =IdY . The existence of such a map j is clearly equivalent to the existence of a completely contractive map ?: Z Ä Y satisfying

(2$) ?bi=IdY , or equivalently

(1$) ? is adjointable and (?~ (x), i( y))A=(x, y)A for all x # Y , y # Y. Condition (2$) above shows that all the above are restatements of the situation of Lemma 7.1(ii). The ? (and hence j) is unique if it exists. If any

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Proposition 7.5. If Y and Z are right A-rigged modules and if (Y, i) is a c-summand of Z, then \: K(Y) Ä K(Z):T[ibTb? is a completely isometric homomorphism. There is a completely contractive retraction map r: K(Z) Ä K(Y) given by T [ ? b T b i. Hence K(Y) may be regarded as a subalgebra of K(Z). Proof. Clearly \ is a homomorphism, \ and r are completely contractive, and r b \=Id which proves the result. K

Note that by Theorem 3.10 we also have \=ihA j. In fact, for the Z of the proposition, K(Z) may be realized as a 2_2 ``linking operator algebra,'' with 11 corner K(Y), and 22 corner K(Z c Y) (see Section 2). Next we point out that by using the same functorial property of the Haagerup tensor product, and Theorem 7.2, we obtain the relations

c c Yk % Z$ (Yk% Z) \ k + k completely isometrically, for any family of right A-rigged modules [Yk], and a right B-rigged module Z, with respect to an essential homomorphism %: A Ä B(Z). We leave out the easy details. Also,

c c Y% Zk $ (Y%k ZK) \ k + k completely isometrically, for a right A-rigged module Y and a family of right B-rigged modules [Zk]. This is with respect to essential c homomorphisms %k : A Ä B(Zk) and %= %k : A Ä B(k Zk) Again we omit the easy proof using Theorem 7.2. Note that one may show that direct sums of such esential homomorphisms is essential using Proposition 6.3:

Lemma 7.6. Let A and B be operator algebras, let [Zk] be a family of right B-rigged modules, and suppose that %k : A Ä B(Zk) is a completely contractive homomorphism for all k. Then %k is essential for all k if and only if c %=k %k : A Ä B(k Zk) is essential.

Remark. Suppose that Yk are rigged modules over an operator algebra A which is represented nondegenerately on a Hilbert space H. Then

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c c c Yk/B H, (Yk hA H ) k \ k + completely isometrically. Explicitly, this inclusion is ( yk)k [

(` [ ( yk `)k). This observation may be viewed as an equivalent definition c of the direct sum k Yk .

Corollary 7.7. (1) Change of rings preserves direct sums: c B B c B (Y1 Ä Y2) $Y 1 Ä Y 2 completely isometrically, for any right A-rigged modules Y1 and Y2 , with respect to an essential homomorphism %: A Ä M(B).

(2) The right rigged module tensor product } hA X with an equivalence bimodule X (as in Theorem 6.1) preserves direct sums completely isometrically.

We define R0(A), the rigged class group, to be the Grothendieck group of the monoid of right A-rigged modules (where addition is the column direct sum Äc), identified up to completely isometric isomorphism. We define R 0(A), the reduced rigged class group,tobeR0(A)ÂH, where H is the subgroup generated by the equivalence class [A].

We also define FR0(A), the K-rigged group, to be the Grothendieck group of the monoid of algebraically finitely generated and projective right A-rigged modules Y, identified up to completely isometric isomorphism. (To see that the last group makes sense one needs to check that a c-direct sum of such modules is again in the class, but this is fairly clear). For an essential homomorphism %: A Ä M(B) and a right A-rigged B module Y, we let R0(%)([Y])=[Y ]. Theorem 7.8. R0 , R 0 , and FR0 are functors from the category OA of operator algebras and essential homomorphisms to the category of abelian groups.

Proof. Note that R0(IdA)=Id. Also R0(% b ?)=R0(%)bR0(?) follows from associativity properties of the Haagerup tensor product. To see that

R0 is a functor we also need to know that change of rings preserves direct sums. However, this was noted above. The statement for FR0 follows immediately. Since R0(%)([A])=[B], the map R0(%) induces a map R 0(%): R 0(A)ÄR 0(B). K It is clear from Theorem 6.1 and Corollary 7.7 above that strongly

Morita equivalent operator algebras have the same R0 and FR0 group.

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We can obviously consider coarser equivalence relations to obtain smaller groups and more interesting invariants, however, the above sets the pattern.

Henceforth let A be unital (recall that K0 of a nonunital algebra is defined by first passing to the unitization). In this case recall that Y =

CBA(Y, A) and K(Y)=CBA(Y) for a finitely generated and projective rigged module Y.

There is a problem in obtaining the algebraic K0 group, namely that we do not believe that every algebraic finitely generated and projective module over an operator algebra may be rigged. This problem may certainly be overcome by using completely bounded maps instead of complete contractions in our definitions. In this case we are dealing with ``completely boundedly-rigged modules,'' or ``cb-rigged modules,'' a topic we intend to pursue more explicitly at a later time. However, in this case the theory is not quite as pretty (for instance our direct sums will only be defined up to a constant). However, by considering finitely generated and projective ``cb-rigged modules'' (these are the cb-rigged modules with K(Y) unital), up to completely bounded isomorphism, we recover the usual K0 group which is well understood. This is because every algebraically finitely generated and projective module over operator algebra may be cb-rigged. Another approach is to consider the class of rigged modules which are a c-summand in Cn(A) for some integer n (this coincides with the notion of a FCGP module defined in Section 8). Note that for C*-algebras, all algebraically finitely generated and projective modules may be realized as such. We can then form a projective class group, IK0(A) with these modules, and essential homomorphisms, but now identifying modules which are algebraically (or, as is fairly clearly seen to be equivalent, completely boundedly) isomorphic, and then the theory that follows mimics the algebraic K-theory quite closely. Indeed IK0(A) is a subgroup of K0(A). Two FCGP right A-modules Y and Z have the same class in IK0(A) if and only if they are completely boundedly finitely stably isomorphic; that is, c c Y1 Ä Cn(A)$Y2 Ä Cn(A) completely boundedly for some n. Note that any such module is a CCGP module (see below), and by Theorem 8.3 is stably isomorphic to a free module. We obtain results such as IK0(AÄ B)=IK0(A)_IK0(B), and so on. As an example, IK0(A(D))=Z if A(D) is the disk algebra. However, it is clear that there are a lot of interesting finitely generated and projective right A(D)-rigged modules which are not in this classsee example 6.9.

8. Countably Generated Rigged Modules and Morita Equivalence

Let W be an essential right A-operator module. Recall that a right

A-module map : C (A) Ä W is finitely A-essential if the restrictions of

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, to Cn(A) are right A-essential. This is equivalent by Proposition 2.5 to w being left multiplication by a fixed element of R (W).

Definition 8.1. We say that a right A-operator module Y is (exactly) countably column generated and approximately projective (CCGP for short) if there are completely contractive right A-module maps

,: Y Ä C (A) and : C (A) Ä Y, with finitely A-essential, such that

b ,=IdY. We say that a right A-operator module Y is (exactly) finitely column generated and approximately projective (FCGP for short) if we can factor as above, but through Cn(A) for some n. Clearly, if Y is FCGP then it is CCGP. From Lemma 7.1 we see that a CCGP-module is automatically a rigged module and the maps , and are adjointable. For a CCGP module Y, by the second half of Proposition 2.5 we t t may write ,( y)=[(x1, y), (x2 , y), ...] and ([a1, a2, ...] )=k=1 ykak, where xk #CBA(Y,A), yk # Y. The condition b ,=IdY implies that k=1 yk(xk, y)=y for all y # Y, where the sum converges in norm (since , maps into C (A)). In [12] the CCGP situation was described as the existence of a (P)-quasi-unit of norm 1. Kasparov's stabilization theorem gives a nice characterization of CCGP modules over a C*-algebra A. We give a quick proof below, which gives just a little more than is usually asserted.

Theorem 8.2. The CCGP modules over a C*-algebra A are precisely the countably generated right A-Hilbert C*-modules. Proof.(o) It follows from a quick calculation (see [12], Section 7) that if Y is a countably generated right A-Hilbert C*-module then B=K(Y) has a strictly positive element. By replacing A with the (C*-subalgebra which is the) closure of the two-sided ideal generated by terms (x| y), for x, y # Y we may assume that the Hilbert C*-module is full. Then A is Morita equivalent to K(Y). Let L be a concrete representation of the linking algebra as a C*-algebra (this allows us to supress the pairings and regard them as concrete operator multiplication: (y|z)=y*z). Following the proof of [18, Lemmas 2.12.3], we can write the identity of the multiplier algebra M(K(Y)) as a sum 1M=k=1 yk yk*, where the sum converges in the strict topology of M(K(Y)). The partial sums of the sum above are uniformly bounded in norm. We now show that t for y # Y we have that ,( y)=[(y1, y), (y2, y), ...] has range in C (A), and that [(y, y1),(y, y2), ...] # R (A). This is equivalent to asking if k=1 (y, yi)( yi, y) converges uniformly, which will follow if we can

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show that k=1 yi(yi, y) converges uniformly to y. Since A is full and A acts nondegenerately on Y, we can assume that y=z(w, x) for x, w, z # Y. Then k=1 yi(yi, y)=k=1 (yi yi*)(zw*)x. By definition of the strict topology this converges to (zw*)x=y. By the second half of Proposition t 2.5, , is completely contractive, as is the map ([a1, a2, ...] )=k=1 ykak from C (A)toY. Clearly we have shown above that ,( y)=y. The other requirements for being CCGP are also clear. ( O ) Suppose, conversely, that Y is a CCGP module over a C*-algebra A. Then since Y is completely isometrically isomorphic to a - submodule of C (A) we see that Y is a Hilbert C* module (has an A- valued inner product). Hence Y =Y*. By the remarks after Definition 8.1, Y is clearly countably generated. K The following result is an absorbtion, or stabilization, theorem.

Theorem 8.3. Suppose that Y is a CCGP module over an operator algebra A.

(1) There exists a c-complemented submodule N in C (A) with c C (A)$YÄ N completely isometrically isomorphically. c (2) C (A)Ä Y$C (A) completely isometrically isomorphically (Y is stably free). c (3) C (A)Ä C (Y)$C (A) completely isometrically isomorphically. Proof. (1) This follows immediately from the discussion in Section 7 on c-complements. (2) We use the ``Eilenberg swindle'':

c c C (A)$C (A)Ä C (A)Ä }}} $(YÄc N)Äc (YÄc N)Äc}}} $YÄc (NÄc Y)Äc (NÄc Y)Äc}}}

c $YÄ C (A).

(3) Follows from (2), since Y is CCGP if and only if C (Y)is CCGP. K We proceed to discuss connections to Morita equivalence. We say that a right A-rigged module Y is full, or that the pairing (}, }):Y _YÄA is full, if the closure of the range of the pairing is all of A and, also, if there is some c.a.i. for A consisting of elements of form n t k=1 (xk, yk), where [x1 , ..., xn] and [ y1 , ..., yn] have norm 1. By

File: 580J 285151 . By:BV . Date:01:02:00 . Time:13:55 LOP8M. V8.0. Page 01:01 Codes: 2803 Signs: 1787 . Length: 45 pic 0 pts, 190 mm 416 DAVID P. BLECHER symmetry, then, Y is a right K(Y)-rigged module. This is all explained in great detail in [12]; however, there is a more functorial ``inductive limit'' version of this condition which should be useful for categorical purposes.

Definition 8.4. Suppose that Y is a right A-rigged module and suppose that there is a net of positive integers m(:) and right A-module maps

_: : A Ä Cm(:)(Y) and {: : Cm(:)(Y) Ä A such that

(1) _: and {: are completely contractive;

(2) {:_: ÄIdA strongly on A;

(3) the maps _: and {: are adjointable;

(4) For all #, _# {: _: Ä _# uniformly. In this case we say that Y is a column generator. We say that a right A-rigged module Y is an (exact) column stable generator (CSG) if there exist completely contractive right A-module maps

_: A Ä C (Y) and {: C (Y) Ä A, with _ b {=IdA. Remark 1. By Lemma 7.1 the maps in the CSG definition are automatically adjointable. It is not hard to see that a CSG rigged module is a column generator. Remark 2. Conditions (1)(4) allow us to regard A as an inductive limit of submodules which factor through spaces Cn(Y).

Remark 3. The maps _: and {: are compact adjointables.

Proposition 8.5. A right A-rigged module is a column generator if and only if it possesses the dual approximate identity property of [12]. In this case Y is an equivalence bimodule for a strong Morita equivalence of A and K(Y).

Proof. Suppose that Y is a column generator. Since _: and {: are compact and adjointable, Theorem 3.10 allows us to realize them explicitly as finite rows or columns, with norms 1. We obtain from condition (2) a left approximate identity [e:] of A, with e: a={: _: a for a # A. This implies that the span of the range of ( }, } ) is dense in A. Let I=[a # A: ae: Ä a], this is a left ideal in A with two-sided c.a.i. (using condition (4)). Indeed IA=A and AI=I, so that A=I as we have seen before. From the above we see that the pairing is full. Thus Y has the dual approximate identity property. The other direction is similar, it uses Theorem 3.10 and Cohen's factorization (to obtain condition (4)). K Thus we see (from Lemma 2.10 in [12]) that if Y is a right A-rigged module which is a column generator, then A is strongly Morita equivalent to K(Y), with equivalence bimodule Y. Conversely, we showed in [12]

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Theorem 8.6. If (A, B, X, Y) is a Morita context, and if Y is CCGP and CSG, then A is completely isometrically isomorphic to B . In par- K K ticular Morita equivalent _-unital C*-algebras are stably isomorphic.

c Proof. We have seen that C (A)$C (Y)Ä C (A). As noted above c C (Y)$C (Y)Ä C (A). Hence C (A)$C (Y). Then, using earlier observations,

A $K(C (A)) K

$K(C (Y)) $K(Y) K $B . K The statement for C*-algebras follows from the above and the argument in [12] showing the existence of a quasi-unit of norm 1 in this case. Such existence implies the CCGP and CSG conditions. K

9. Index for Subalgebras of Operator Algebras

It has been made clear (very explicitly in [58]), that at the heart of the Jones index for subfactors is a purely ring theoretic construction. There- fore, at an algebraic level it is quite obvious how to define index for a unital subalgebra A of a (possibly non-self-adjoint) unital algebra B, when there exists an algebraic conditional expectation E: B Ä A. This is a surjective A-bimodule map E with E(a)=a for all a # A. Motivated by the work of Pimsner and Popa [46] on finite Jones index, Watatani defines a

``quasi-basis'' for a conditional expectation. This is a finite family [(u1 , v1),

(u2 , v2), ..., (un , vn)]/B_B with k uk E(vk b)=b=k E(buk)vk , for all b # B. A conditional expectation which possesses a quasi-basis is said to be of index-finite type. From these purely algebraic notions, one proceeds to the algebraic basic construction, and a substantial theory develops. However, in the self-adjoint theory, in order to obtain the interesting analytic consequences of the index, it is necessary to do all the above in a C*-algebraic context. That is we need the ``C*-basic construction,'' which

File: 580J 285153 . By:BV . Date:01:02:00 . Time:13:55 LOP8M. V8.0. Page 01:01 Codes: 2951 Signs: 2105 . Length: 45 pic 0 pts, 190 mm 418 DAVID P. BLECHER essentially uses Hilbert C*-modules. We outline very briefly how one may use the ideas contained in this paper (and [12]) to extend the analytic index theory to (possibly nonselfadjoint) operator algebras. Again we concentrate on the ``1+='' version of our theory, and we assume that the operator algebras A and B are unital, with the common identity of norm 1, for simplicity.

Definition 9.1. Let A be a unital subalgebra of a unital operator algebra B, and suppose that E: B Ä A is a completely contractive unital conditional expectation. We say that E is of 1-index-finite type if, given ======>0, there exists a quasi-basis [(u 1 , v1), (u2 , v2), ..., (u n(=), v n(=))]/B_B such that:

= =$ (1) &[E(vi bkl u j )](i, k), ( j, l)&<(1+=)(1+=$) &[bkl]&, for all =, =$>0, and for all finite matrices [bkl] with entries in B.

= = t (2) &[E(v1), ..., E(vn(=))] &<(1+=), for all =>0.

= = (3) &[E(u1), ..., E(u n(=))]&<(1+=), for all =>0. We have not tried particularly hard to eliminate redundancies in the list above. To justify these conditions we note that in the C*-algebra case, by

Lemma 2.1.6 of [58], one can always take ==0, and uk=vk*, and then conditions (1)(3) follow (we omit the proof of these). If we insist on ==0 in the non-self-adjoint case, then we expect to obtain a particularly nice version of the theory, but one which is probably too restrictive in general.

In the situation described above, let Y0=B, and define ,=(b)= = = t t = [E(v1b), ..., E(v n(=) b)] and =([a1, ..., an(=)] )=k ukak. Then ,= : Y0 Ä Cn(=)(A), = : Cn(=)(A) ÄY0 are right A-module maps. Moreover, the conditions of Definition 4.1 essentially hold, so that we can induce a right

A-rigged module structure on Y0 . In fact, we define matrix norms on Y0 by

= &[bij]&=lim sup &[E(vk bij)](i, k), j &. = Ä 0

An elementary argument shows that these matrix norms are uniformly equivalent to the original matrix norms on B. We let Y be the usual

A-rigged module which is the completion of Y0 (in this case Y=Y0). A symmetric construction yields Y , which is simply B regarded as a left A-module, and which is normed by

= &[bij]&=lim sup &[E(bij u k)]i,(j,k)&. = Ä 0

The A-valued pairing of Y with Y is (x, y)A=E(xy).

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Form B(Y) as in Section 3 and note that this equals K(Y), since the latter has identity. Define a map *: B Ä B(Y) by the left regular representation of B and Y; this turns out to be a completely contractive homomorphism. We define eA to be E viewed as an element of B(Y); this turns out to be a completely contractive, adjointable (self-adjoint), projection. We obtain the

Jones relation eA*(b) eA=*(E(b)) eA for b # B. As in Section 3 we have that B(Y)=YhA Y is an operator algebra. We rename this operator algebra OA(B, eA), and call it the operator algebra basic construction. The conditions above also give that A is strongly Morita equivalent (in the sense of [12]) to OA(B, eA), with equivalence bimodule Y. We remark that the ideas above and Definition 4.1 show how to proceed in the non-index-finite case. = = We define Index E to be the element k u kv k of B. The usual argument shows that this is independent of the particular choice of quasi-basis and lies in the center of B.

As observed above, B is embedded in OA(B, eA) by the map *. Assum- ing that Index E is invertible, one may then define a conditional expectation of OA(B, eA) onto B by | y)(x|=yeA x [ (1ÂIndex E) yx. One may also construct compositions, direct sums, and tensor products of conditional expectations as described in [58], using the methods of Section 6 above. We are in the process of looking for nice examples; until we have found such we feel that we are not justified in taking our theory any further at this point.

To illustrate the above in a trivial case, consider B=M2(A(D)), A=A(D) (the disk algebra), and let E: B Ä A be the map

a b [ ta+(1&t)d. _c d& Here 0t1. The quasibasis of Example 1.2.5 of [58] satisfies the condition of Definition 9.1, so that E is of index-finite type, and Index E=(1Ât+1Â(1&t))1. This value lies in [4, ).

References

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