Stably isomorphic dual operator algebras

Trans. Amer. Math. Soc.

2011

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.

Section: Theorem 28 If Y Is a Right W * -Module Over M And If Z Imentioning

confidence: 90%

“…[10,Chapter 4]) plays a considerable role in our paper. Indeed the absence of this tool, and also of a recently introduced module tensor product [17] (see also [8,Section 2]) is the main reason why headway was not made on this project many years ago.…”

Section: Introduction and Notationmentioning

confidence: 99%

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

Blecher

Trans. Amer. Math. Soc.

2011

“…Composing the map M ⊗ σh (X ⊗ σh Y) → X ⊗ σh Y above with the canonical map M × (X ⊗ σh Y) → M ⊗ σh (X ⊗ σh Y), one sees the action of M on X ⊗ σh Y is separately weak* continuous (see also [20]). That (a 1 a 2 )z = a 1 (a 2 z) for a i ∈ M, z ∈ X ⊗ σh Y, follows from the weak* density of X ⊗ Y, and since this relation is true if z is finite rank.…”

Section: Some Tensor Productsmentioning

confidence: 95%

“…We define X ⊗ σh M Y ⊗ σh N Z to be the quotient of X ⊗ σh Y ⊗ σh Z by the w * -closure of the linear span of terms of the form xm ⊗ y ⊗ z − x ⊗ my ⊗ z and x ⊗ yn ⊗ z − x ⊗ y ⊗ nz, with x ∈ X, y ∈ Y, z ∈ Z, m ∈ M, n ∈ N. By extending the arguments of Proposition 2.2 in [20] to the threefold normal module Haagerup tensor product, one sees that X ⊗ σh M Y ⊗ σh N Z has the following universal property: if W is a dual operator space and u : X × Y × Z → W is a separately w * -continuous, completely contractive, balanced, trilinear map, then there exists a w * -continuous and completely contractive, linear map…”

Section: Some Tensor Productsmentioning

confidence: 98%

Morita equivalence of dual operator algebras

Blecher

Journal of Pure and Applied Algebra

2008

108

Communicated by C. Kassel MSC: Primary: 47L30 47L45 46L08 secondary: 16D90 47L25a b s t r a c tWe consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel's W * -algebraic Morita equivalence.Published by Elsevier B.V.

“…In [16] it was shown that the -equivalence implies weak * Morita equivalence in the sense of [4]. That is, any of the equivalences of [15] is one of our weak * Morita equivalences.…”

Section: Introduction and Notationmentioning

confidence: 94%

A Morita theorem for dual operator algebras

Journal of Functional Analysis

2009

We prove that two dual operator algebras are weak * Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401-2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak * -continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak * Morita equivalence bimodule. We also develop the theory of the W * -dilation, which connects the non-selfadjoint dual operator algebra with the W * -algebraic framework. In the case of weak * Morita equivalence, this W * -dilation is a W * -module over a von Neumann algebra generated by the nonselfadjoint dual operator algebra. The theory of the W * -dilation is a key part of the proof of our main theorem. Published by Elsevier Inc.