Duality and operator algebras: automatic weak* continuity and applications

Abstract

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak * -continuous on dual spaces. In particular, if X is a subspace of a C * -algebra A, and if a ∈ A satisfies aX ⊂ X, then we show that the function x → ax on X is automatically weak * continuous if either (a) X is a dual operator space, or (b) a * X ⊂ X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C * -subalgebra. Applications include a new characterization of the -weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W * -modules to the framework of modules over such algebras. We also give a Banach module characterization of -weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.

Dual Operator Algebras with Normal Virtual h-Diagonal

“…Actually this w * -continuity is automatic from Corollary 4.10 in [3]. Consequently, from the universal property of the normal Haagerup tensor product, X is a normal operator M-module if and only if…”

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

“…Such spaces can be represented completely isometrically as concrete dual operator bimodules, and in fact this can be done under even weaker hypotheses (see e.g. [10,11,15]) and similarly for one-sided modules (the case M or N equals C). We use standard notation for module mapping spaces; e.g.…”

“…[23,14,11]). These objects were first studied by Paschke, and then by Rieffel [21,22] (see also e.g.…”

“…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.…”

Duality and operator algebras II: operator algebras as Banach algebras

“…This rules out the possibility, which had remained open, of a 'nonoperator-space variant' of the following theorem attributable to Le Merdy and the two authors (e.g. see [3,Section 2.7,6,2]): namely that the -weakly closed operator algebras 'are precisely' the operator algebras which possess an operator space predual. Thus, we are able to bring to its final form the topic of abstract characterizations of -weakly closed operator algebras.…”