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Categories of operator modules (Morita equivalence and projective modules). (English) Zbl 0966.46033

Mem. Am. Math. Soc. 681, 94 p. (2000).
This article surveys work in progress on the generalisation to not necessarily self-adjoint operator algebras of M. A. Rieffel’s work [J. Pure Appl. Alg. 5, 51-96 (1974; Zbl 0295.46099)] on Morita equivalence for \(W^{*}\) and \(C^{*}\)- algebras. This in turn was a generalisation of Morita ring theory [K. Morita, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83-142 (1958; Zbl 0080.25702)]. The main aim of Morita equivalence is to investigate the relation between algebraic objects and their representation or module theory. A Morita context, or set of equivalence data, between unital rings \(R\) and \( S\) is a system \((R,S,P,Q,(\cdot,\cdot), [\cdot,\cdot ])\) where \(P\) is a projective module, \(Q\) is isomorphic to \( \operatorname{Hom}_{R}(P,R)\) and the two pairings, here bimodule homomorphisms, take values in \(R\) and \(S\) respectively. The rings are called Morita equivalent if they can be embedded in a Morita context. In general Morita equivalence is coarser than isomorphism. Algebraically, Morita equivalence is the same as the equivalence of the categories of left modules (or the categories of representations). When the algebras are given additional structure one needs to adjust the concept of module and introduce a “strong” Morita equivalence. In general Morita equivalence is coarser than isomorphism. The background to the \(C^{*}\)-algebraic theory comes from the representation theory for full hereditary sub-\(C^{*}\)-algebras (the extension of their pure states) and their modules, from Rieffel’s “rigged” modules and systems of imprimivity, and from representations of locally compact groups induced from representations of a subgroup. Indeed, G. Mackey’s imprimitivity theorem gives a Morita equivalence. Morita equivalence of operator algebras \(A\) and \(B\) entails adapting structures such as module, tensor product and classes of mappings to be appropriate to the type of algebra in question.
The authors consider “operator spaces”, i.e., modules over operator algebras, and metricize (using either completely isometries or completely bounded isomorphisms) Rieffel’s Hilbert \(C^{*}\)-modules to general operator algebras. Strongly (as in Riefel) and completely bounded Morita equivalences have different applications. For a Morita context the authors need to show completely contractive equivalence of the categories and they require an associativity condition relating the two pairings, as in the classical theory, and they also require “P” (for projective) and “G” (for generator) conditions so that the pairings are complete quotient maps to \(A\) and to \(B\) respectively. Morita equivalent algebras always come with a pair of corresponding bimodules in such a way that equivalence of categories corresponds to tensoring of these bimodules.
Let \(X\) and \(Y\) respectively be operator \(A,B\)- and \(B,A\)-bimodules and consider, for instance, \( Y \otimes_{hB} X\), the “module” Haagerup tensor product of operator algebras [see V. I. Paulsen and R. R. Smith, J. Funct. Anal. 73, 258-276 (1987; Zbl 0644.46037)]. This is self dual, being both injective and projective. The operator space projective tensor product norm, which is usually suitable when dealing with categories is not suitable in the authors’ context. The Haagerup tensor product, in the same way that a tensor product linearises bilinear mappings, linearises operator multiplication. The mapping \((\cdot)_{A}: {A \times B} \to A\), for a completely contractive \(A\)-valued “inner-product” \(( \cdot, \cdot)\) on \(X \times Y\), induces an isometry from \({X \otimes Y} / \text{Ker} (\cdot)_{A}\) onto \(Y\) if and only if there is a majorisation for the norms of tensors of finite rank. This majorisation is used subsequently for a dual “approximate identity” property.
Inn Chapter IV the authors develop a concept of “projective” operator module but this does not fit in categorically with the ring-theoretic projective module. The ring-theoretic projective module can be defined as a direct summand of a free module, but the direct sums of operator modules are not well-defined. For C\(^{*}\)-algebras the Hilbert C\(^{*}\)-modules are suitable as projective modules [see D. P. Blecher, Math. Ann. 307, No. 2, 253-290 (1997; Zbl 0874.46037)]. The definition of a left (P)-module relates to the dual approximate identity mentioned above. Duality relates to equivalence of right A-modules and left B-modules. In ring theory this can be understood as a reversal of arrows (contravariance as well as covariance) in the diagram for a projective module. The authors need complete contractivity and the dual approximate identity. As the space of endomorphisms is seldom an algebra, unlike in the classical algebraic theory, and it is an unsuitable object for duality of operator bimodules. It is replaced by an ideal \(\text{end}(X)\) of the algebra of ‘adjointable’ operators. (For rigged left \(A\)-modules \(Y\) and \(Z\) and corresponding right \(A\)-modules \({\widetilde{Y}}\) and \({\widetilde{Z}}\) the operator \(T: Y \to Z\) is said to be adjointable if there exists an \({\widetilde{T}}\) such that \((z,Ty) = ({\widetilde{T}}z,y)\) for \(y \in Y, z \in {\widetilde{Z}}\)).
In Chapter V the authors adapt the linking algebra \({\mathcal{L}}\) to be an operator algebra of \(4 \times 4\) matrices with coefficients in \(A\), \(B\), \(X\) and \(Y\), such that the embeddings of these in \({\mathcal{L}}\) are completely bounded. It is called linking as, for example, given \(X\) and \(B\) one may use it to recapture \(A\).
Chapter VI and VII give a comparison with Rieffel theory and the new approach is used to get improved results in the \(C^{*}\) theory. For instance they extend the \(C^*\) stable isomorphism theorem to operator algebras which have what they call a quasi-unit of norm 1; this imples for instance that if \(A \otimes K\) and \(B \otimes K\) are algebraically and completely isometrically equivalent, \(K\) denoting the \(C^{*}\)-algebra of compact operators on a separable Hilbert space, then \(A\) and \(B\) are strongly Morita equivalent.
Chapter VIII constitutes examples. They consider operator algebras which are infinite generalisations of incidence algebras and operator algebras of groupoids (cf. G. Mackey’s virtual groups). Examples in complex analysis illustrate differences between completely bounded and strong Morita equivalence. There is also an appendix on progress subsequent to the preparation of the memoir.

MSC:

46L07 Operator spaces and completely bounded maps
46M10 Projective and injective objects in functional analysis
46L08 \(C^*\)-modules
47L30 Abstract operator algebras on Hilbert spaces
47L55 Representations of (nonselfadjoint) operator algebras
47L25 Operator spaces (= matricially normed spaces)
16D90 Module categories in associative algebras
46L06 Tensor products of \(C^*\)-algebras
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