On the reflexivity properties of Banach bundles and Banach modules

Source abstract

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