Reflexivity of Banach $C(K)$-modules via the reflexivity of Banach lattices

TL;DR

This paper extends criteria for reflexivity from Banach lattices to finitely generated Banach $C(K)$-modules, showing they are reflexive if they exclude subspaces isomorphic to $l^1$ or $c_0$, thus broadening understanding of module reflexivity.

Contribution

It introduces a reflexivity criterion for finitely generated Banach $C(K)$-modules based on subspace exclusion, generalizing known lattice results.

Findings

Finitely generated Banach $C(K)$-modules are reflexive iff they lack $l^1$ or $c_0$ subspaces.

Extension of Lozanovsky and Lotz criteria to Banach $C(K)$-modules.

Provides a characterization of reflexivity in this broader class.

Abstract

We extend the well known criteria of reflexivity of Banach lattices due to Lozanovsky and Lotz to the class of finitely generated Banach $C(K)$- modules. Namely we prove that a finitely generated Banach $C(K)$-module is reflexive if and only if it does not contain any subspace isomorphic to either $l^1$ or $c_0$.