Categories of operator modules (Morita equivalence and projective modules)

Blecher, David P.; Muhly, Paul S.; Paulsen, Vern I.

doi:10.1090/memo/0681

Cited by 70 publications

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“…Remark 1. The one direction of the``if and only if'' of the theorem is easy and was noted in [9]. For completeness we sketch the short argument here.…”

Section: Notation Background and Statement Of The Theoremmentioning

confidence: 89%

On Morita's fundamental theorem for $C^*$-algebras

Blecher¹

2001

MATH. SCAND.

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We give a solution, via operator spaces, of an old problem in the Morita equivalence of C*-algebras. Namely, we show that C*-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a C Ã -algebra a is a closed subspace of some B(H) which is left invariant under multiplication by % a , where % is a *-representation of a on H. The category a HMOD of *-representations of a on Hilbert space is a full subcategory of the category a OMOD of operator modules. Our main result remains true with respect to subcategories of OMOD which contain HMOD and the C Ã -algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a C Ã -algebra argument using compactness of the quasistate space of a C Ã -algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

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“…Remark 1. The one direction of the``if and only if'' of the theorem is easy and was noted in [9]. For completeness we sketch the short argument here.…”

Section: Notation Background and Statement Of The Theoremmentioning

confidence: 89%

On Morita's fundamental theorem for $C^*$-algebras

Blecher¹

2001

MATH. SCAND.

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“…The following is the 'weak* topology variant' of the notion of rigged module studied in [2,12,3,6] (the last paper has the most succinct definition of these objects, and [3] is a survey). In Section 4 we will prove several equivalent, but quite different looking, characterizations of w * -rigged modules.…”

Section: 2])mentioning

confidence: 99%

“…These are the appropriate weak* topology variant of our earlier notion of rigged modules (see e.g. [2,12,3,6]), which in turn generalizes the notions of Hilbert C * -modules and Hilbert space, successively. W * -rigged modules will have an extensive theory; we will only present the basics here.…”

Section: Introduction and Notationmentioning

confidence: 99%

“…Since a number of the ideas and proofs here are quite analogous to those from our papers on related topics, principally [2,12,8,11], we will be quite brief in many of the proofs. We assume that the reader has some familiarity with these earlier ideas and proof techniques, and will often merely indicate the new techniques which allow one to modify arguments from [2,12] to the present (weak* topology) setting.…”

Section: Introduction and Notationmentioning

confidence: 99%

“…We assume that the reader has some familiarity with these earlier ideas and proof techniques, and will often merely indicate the new techniques which allow one to modify arguments from [2,12] to the present (weak* topology) setting. As the paper proceeds we will include fewer and fewer details, since we will be assuming that the reader is growing in familiarity with the methods introduced earlier and in [8], and the ways these are used to modify proofs from the older theory.…”

Section: Introduction and Notationmentioning

confidence: 99%

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A characterization and a generalization of $W^{*}$-modules

Blecher

Kashyap

2011

Trans. Amer. Math. Soc.

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Abstract. We give a new Banach module characterization of W * -modules, also known as self-dual Hilbert C * -modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W * -modules, to the setting where the operator algebras are σ-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak* topology variant of our earlier notion of rigged modules, and their theory, which in turn generalizes the notions of a C * -module and a Hilbert space, successively. Our w * -rigged modules have canonical 'envelopes' which are W * -modules. Indeed, a w * -rigged module may be defined to be a subspace of a W * -module possessing certain properties.

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A Topology for Operator Modules overW*-Algebras

Magajna

1998

Journal of Functional Analysis

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dedicated to professor ivan vidav in honor of his eightieth birthday Given a von Neumann algebra R on a Hilbert space H, the so-called R-topology is introduced into B(H), which is weaker than the norm and stronger than the ultrastrong operator topology. A right R-submodule X of B(H ) is closed in the R-topology if and only if for each b # B(H ) the right ideal, consisting of all a # R such that ba # X, is weak* closed in R. Equivalently, X is closed in the R-topology if and only if for each b # B(H ) and each orthogonal family of projections e i in R with the sum 1 the condition be i # X for all i implies that b # X. Academic Press

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Categories of operator modules (Morita equivalence and projective modules)

Cited by 70 publications

References 0 publications

On Morita's fundamental theorem for $C^*$-algebras

On Morita's fundamental theorem for $C^*$-algebras

A characterization and a generalization of $W^{*}$-modules

A Topology for Operator Modules overW*-Algebras

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