Small Banach bundles and modules

TL;DR

This paper characterizes Banach bundles over compact Hausdorff spaces whose sections are finitely generated modules over the function algebra, extending previous work and providing equivalent conditions involving local triviality and decomposability.

Contribution

It provides a comprehensive characterization of Banach bundles with finitely generated sections, answering a specific open question and extending results to non-metrizable base spaces.

Findings

Characterization of Banach bundles with finitely generated sections.

Equivalence of conditions involving local triviality and decomposability.

Extension of previous results to broader classes of base spaces.

Abstract

We characterize those (continuously-normed) Banach bundles $\mathcal{E}\to X$ with compact Hausdorff base whose spaces $\Gamma(\mathcal{E})$ of global continuous sections are topologically finitely-generated over the function algebra $C(X)$, answering a question of I. Gogi\'c's and extending analogous work for metrizable $X$. Conditions equivalent to topological finite generation include: (a) the requirement that $\mathcal{E}$ be locally trivial and of finite type along locally closed and relatively $F_{\sigma}$ strata in a finite stratification of $X$; (b) the decomposability of arbitrary elements in $\ell^p(\Gamma(\mathcal{E}))$, $1\le p<\infty$ as sums of $\le N$ products in $\ell^p(C(X))\cdot \Gamma(\mathcal{E})$ for some fixed $N$; (c) the analogous decomposability requirement for maximal Banach-module tensor products $F\widehat{\otimes}_{C(X)}\Gamma(\mathcal{E})$ or (d) equivalently, only for $F=\ell^1(C(X))$.