The dual Radon–Nikodym property for finitely generated Banach C(K)-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 $$\ell ^1$$ .