Hilbert $C^*$-modules over $\Sigma^*$-algebras

TL;DR

This paper develops a theory of Hilbert modules over $\Sigma^*$-algebras, extending key results from $C^*$- and $W^*$-modules and introducing a $\Sigma^*$-module completion with unique properties.

Contribution

It introduces $\Sigma^*$-modules, extending the theory of $C^*$- and $W^*$-modules, and defines a $\Sigma^*$-module completion with a uniqueness property.

Findings

Established $\Sigma^*$-versions of $C^*$- and $W^*$-module results.

Developed a $\Sigma^*$-module completion with a uniqueness condition.

Analyzed modules with weak sequential countable generation.

Abstract

A $\Sigma^*$-algebra is a concrete $C^*$-algebra that is sequentially closed in the weak operator topology. We study an appropriate class of $C^*$-modules over $\Sigma^*$-algebras analogous to the class of $W^*$-modules (selfdual $C^*$-modules over $W^*$-algebras), and we are able to obtain $\Sigma^*$-versions of virtually all the results in the basic theory of $C^*$- and $W^*$-modules. In the second half of the paper, we study modules possessing a weak sequential form of the condition of being countably generated. A particular highlight of the paper is the "$\Sigma^*$-module completion," a $\Sigma^*$-analogue of the selfdual completion of a $C^*$-module over a $W^*$-algebra, which has an elegant uniqueness condition in the countably generated case.