Non-commutative branched covers and bundle unitarizability

TL;DR

The paper investigates when Banach bundles over a base space can be renormed to Hilbert bundles, equivalently when the section space $\Gamma({\mathcal A})$ admits a finite-index conditional expectation onto $C(X)$. It proves a positive result: for a unital subhomogeneous $(F)$ $C^*$-bundle over a compact metrizable base, the embedding $C_b(X)\hookrightarrow \Gamma_b({\mathcal A})$ admits a finite-index expectation, but it simultaneously shows that the optimal index $K(E)=r({\mathcal A})$ is not generally achievable, via a concrete counterexample with $r({\mathcal A})=3$. It further demonstrates that homogeneous $n$-homogeneous Banach bundles over locally paracompact bases are boundedly unitarizable, with a quantified Banach-Mazur-distance bound to a Hilbert-bundle renorming provided by fiberwise Löwner ellipsoids that vary continuously in $x$ (Vietoris topology). The work combines Bratteli-germ analysis and Michael-type selection techniques to construct finite-index expectations with uniform $K(E)$ bounds, clarifying when optimal indices can be realized and offering a quantitative framework for unitarizability in the metrizable setting. The results advance the understanding of non-commutative branched covers and the stability of Hilbert-module structures under renorming.

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-Gogić (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 Gogić.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Definition 1.1
• Lemma 1.2
• Proof 1
• Proposition 1.3
• Proof 2
• Remark 1.4
• Corollary 1.5
• Proof 3