On the reflexivity properties of Banach bundles and Banach modules

Abstract

In this paper, we investigate some reflexivity-type properties of separable measurable Banach bundles over a σ-finite measure space. Our two main results are the following: • The fibers of a bundle are uniformly convex (with a common modulus of convexity) if and only if the space of its L p-sections is uniformly convex for every p ∈ (1,∞). • The fibers of a bundle are reflexive if and only if the space of its L p-sections is reflexive for every p ∈ (1,∞). They generalise well-known results for Lebesgue–Bochner spaces.