Non-commutative branched covers and bundle unitarizability

TL;DR

This paper investigates the structure of continuous $C^*$-bundles and Banach bundles over topological spaces, establishing finite-index expectations, limitations on optimal index, and conditions for renorming into Hilbert bundles, thus addressing several open questions.

Contribution

It proves the existence of finite-index expectations for certain $C^*$-bundles, shows such expectations lack an optimal index in general, and demonstrates how Banach bundles can be renormed into Hilbert bundles.

Findings

Existence of finite-index expectations for subhomogeneous $C^*$-bundles.

Negative answer to the existence of optimal index for expectations.

Banach bundles can be renormed into Hilbert bundles close to the original in Banach-Mazur sense.

Abstract

We prove that (a) the sections space of a continuous unital subhomogeneous $C^*$ bundle over compact metrizable $X$ admits a finite-index expectation onto $C(X)$, answering a question of Blanchard-Gogi\'{c} (in the metrizable case); (b) such expectations cannot, generally, have ``optimal index'', answering negatively a variant of the same question; and (c) a homogeneous continuous Banach bundle over a locally paracompact base space $X$ can be renormed into a Hilbert bundle in such a manner that the original space of bounded sections is $C_b(X)$-linearly Banach-Mazur-close to the resulting Hilbert module over the algebra $C_b(X)$ of continuous bounded functions on $X$. This last result resolves quantitatively another problem posed by Gogi\'{c}.