Module weak Banach-Saks and module Schur properties of Hilbert C*-modules

TL;DR

This paper explores module versions of Banach-Saks and Schur properties within Hilbert C*-modules, providing new characterizations and relations to self-duality, extending classical Banach space theory.

Contribution

It introduces and studies natural module analogues of Banach-Saks and Schur properties in Hilbert C*-modules, including characterizations and their relation to self-duality.

Findings

Characterizations of weak module topologies

Module Banach-Saks and Schur properties defined and analyzed

Relations established between these properties and self-duality

Abstract

Continuing the research on the Banach-Saks and Schur properties started in (cf. M. Frank, A. A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert C*-modules (submitted)) we investigate analogous properties in the module context. As an environment serves the class of Hilbert C*-modules. Some properties of weak module topologies on Hilbert C*-modules are described. Natural module analogues of the classical weak Banach-Saks and the classical Schur properties are defined and studied. A number of useful characterizations of properties of Hilbert C*-modules is obtained. In particular, some interrelations of these properties with the self-duality property of countably generated Hilbert C*-modules are established.