Weak Sequential Completeness in Banach $C(K)$-modules of finite multiplicity

TL;DR

This paper extends Lozanovsky's characterization of weak sequential completeness from Banach lattices to Banach $C(K)$-modules of finite multiplicity, showing the property depends on cyclic subspaces.

Contribution

It generalizes the concept of weak sequential completeness to a broader class of Banach modules, linking it to cyclic subspace properties.

Findings

Banach $C(K)$-modules of finite multiplicity are weakly sequentially complete iff all cyclic subspaces are.

Extension of Lozanovsky's theorem to Banach $C(K)$-modules.

Characterization of weak sequential completeness in finitely generated Banach $C(K)$-modules.

Abstract

A well known result of Lozanovsky states that a Banach lattice is weakly sequentially complete if and only if it does not contain a copy of $c_{0}$. In the current paper we extend this result to the class of Banach $C(K)$ modules of finite multiplicity and, as a special case, to finitely generated Banach $C(K)$-modules. Moreover, we prove that such a module is weakly sequentially complete if and only if each cyclic subspace of the module is weakly sequentially complete.