Symmetric monoidal categories of conveniently-constructible Banach bundles

Abstract

We show that a continuously-normed Banach bundle $E$ over a compact Hausdorff space X whose space of sections is algebraically finitely-generated (f.g.) over $C\left(X\right)$ is locally trivial (and hence the section space is projective f.g over $C\left(X\right)$); this answers a question of I. Gogić. As a preliminary we also provide sufficient conditions for a quotient bundle to be continuous phrased in terms of the Vietoris continuity of the unit-ball maps attached to the bundles. Related results include (a) the fact that the category of topologically f.g. continuous Banach bundles over X is symmetric monoidal under the (fiber-wise-maximal) tensor product, (b) the full faithfulness of the global-section functor from topologically f.g. continuous bundles to $C\left(X\right)$-modules and (c) the consequent identification of the algebraically f.g. bundles as precisely the rigid objects in the aforementioned symmetric monoidal category.