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A Generalization of Hilbert Modules Journal of Functional Analysis 2851 journal of functional analysis 136, 365421 (1996) article no. 0034 A Generalization of Hilbert Modules 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. * Supported by a grant from the NSF. E-mail: dblecherÄmath.uh.edu. 365 0022-1236Â96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 580J 285101 . By:BV . Date:01:02:00 . Time:13:54 LOP8M. V8.0. Page 01:01 Codes: 3360 Signs: 1787 . Length: 50 pic 3 pts, 212 mm 366 DAVID P. BLECHER 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 else- where. 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 dimen- sion 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 File: 580J 285102 . By:BV . Date:01:02:00 . Time:13:54 LOP8M. V8.0. Page 01:01 Codes: 3244 Signs: 2697 . Length: 45 pic 0 pts, 190 mm GENERALIZATION OF HILBERT MODULES 367 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 com- pletely 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 con- verse 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 discus- sion 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]).
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