A characterization and a generalization of W*-modules

TL;DR

This paper extends the concept of W*-modules to a broader setting using a new Banach module characterization, introducing w*-rigged modules that generalize C*-modules and Hilbert spaces with canonical W*-module envelopes.

Contribution

It provides a new Banach module characterization of W*-modules and generalizes the theory to σ-weakly closed operator algebras, defining w*-rigged modules with W*-module envelopes.

Findings

Introduces a new Banach module characterization of W*-modules.

Defines w*-rigged modules as subspaces of W*-modules with specific properties.

Establishes that w*-rigged modules have canonical W*-module envelopes.

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 $\sigma$-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.