A characterization and a generalization of W*-modules

Details A characterization and a generalization of W*-modules A characterization and a generalization of W*-modules

2007-12-09

arxiv.org/abs/0712.1236v2

Mathematics

Operator Algebras

19 pages

Scientific paper

We give a new Banach module characterization of $W^*$-modules, also known as selfdual 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 {\em rigged modules}, and their theory, which in turn generalizes the notions of C*-module, and Hilbert space, successively. Our {\em w*-rigged modules} have canonical `envelopes' which are W*-modules. Indeed, w*-rigged modules may be defined to be a subspace of a W*-module possessing certain properties.

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