On the reflexivity properties of Banach bundles and Banach modules

TL;DR

This paper explores the reflexivity and convexity properties of Banach bundles and their associated $L^p$-spaces, establishing equivalences that generalize known results for Lebesgue-Bochner spaces.

Contribution

It provides new characterizations linking fiber properties of Banach bundles to the reflexivity and convexity of their $L^p$-sections, extending classical results.

Findings

Fibers are uniformly convex iff $L^p$-sections are uniformly convex for all $p$

Fibers are reflexive iff $L^p$-sections are reflexive

Generalizes classical results for Lebesgue-Bochner spaces

Abstract

In this paper we investigate some reflexivity-type properties of separable measurable Banach bundles over a $\sigma$-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 $p\in(1,\infty)$. - The fibers of a bundle are reflexive if and only if the space of its $L^p$-sections is reflexive. These results generalise the well-known corresponding ones for Lebesgue-Bochner spaces.