The dual Radon - Nikodym property for finitely generated Banach C(K)-Modules

TL;DR

This paper extends Lotz's criterion for the Radon-Nikodym property to finitely generated Banach $C(K)$-modules, showing the dual has RNP iff the space lacks an $ ext{ell}^1$ subspace.

Contribution

It generalizes the RNP criterion from Banach lattices to finitely generated Banach $C(K)$-modules and modules of finite multiplicity.

Findings

Dual RNP holds iff no $ ext{ell}^1$ subspace exists.

Extends Lotz's criterion to broader classes of Banach modules.

Provides a characterization of RNP in these modules.

Abstract

We extend the well-known criterion of Lotz for the dual Radon-Nikodym property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules and Banach $C(K)$-modules of finite multiplicity. Namely, we prove that if $X$ is a Banach space from one of these classes then its Banach dual $X^\star$ has the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell^1$.