A characterization and a generalization of W*-modules

Abstract

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 $σ$-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.