A characterization and a generalization of 𝑊*-modules

Blecher, David; Kashyap, Upasana

doi:10.1090/S0002-9947-2010-05153-0

A characterization and a generalization of $W^*$-modules
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by David P. Blecher and Upasana Kashyap PDF

Trans. Amer. Math. Soc. 363 (2011), 345-363 Request permission

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 $\sigma$-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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