Single and multi-valued Hilbert-bundle renormings

TL;DR

This paper characterizes subhomogeneous Banach bundles over compact metrizable spaces as equivalent to Hilbert bundles and explores related structures, extending known results and providing new insights into their properties.

Contribution

It proves the equivalence of subhomogeneous Banach bundles to Hilbert bundles over compact metrizable spaces and extends results to $C^*$ bundles with finite-index expectations.

Findings

Subhomogeneous Banach bundles are equivalent to Hilbert bundles over compact metrizable spaces.

Metrizability is essential for the equivalence; without it, the result does not hold.

Subhomogeneous $C^*$ bundles admit finite-index expectations.

Abstract

We prove that subhomogeneous continuous Banach bundles over compact metrizable spaces are equivalent to Hilbert bundles, while examples show that the metrizability assumption cannot be dropped completely. This complements the parallel statement for homogeneous bundles without the metrizability assumption, and generalizes the analogous result to the effect that subhomogeneous $C^*$ bundles over compact metrizable spaces admit finite-index expectations.