Equivariant Banach-bundle germs

TL;DR

This paper proves that homogeneous equivariant subbundles of Banach or Hilbert space bundles over paracompact spaces extend from invariant closed subsets to neighborhoods under group actions, generalizing previous results.

Contribution

It extends the theory of equivariant bundle extensions to Banach and $C^*$-algebra bundles, including new results on local extensibility and approximability.

Findings

Homogeneous equivariant subbundles extend from invariant closed subsets to neighborhoods.

Equivariant bundle classifying spaces are absolute neighborhood extensors.

Almost-multiplicative maps can be approximated by Banach algebra morphisms.

Abstract

Consider a continuous bundle $\mathcal{E}\to X$ of Banach/Hilbert spaces or Banach/$C^*$-algebras over a paracompact base space, equivariant for a compact Lie group $\mathbb{U}$ operating on all structures involved. We prove that in all cases homogeneous equivariant subbundles extend equivariantly from $\mathbb{U}$-invariant closed subsets of $X$ to closed invariant neighborhoods thereof (provided the fibers are semisimple in the Banach-algebra variant). This extends a number of results in the literature (due to Fell for non-equivariant local extensibility around a single point for $C^*$-algebras and the author for semisimple Banach algebras). The proofs are based in part on auxiliary results on (a) the extensibility of equivariant compact-Lie-group principal bundles locally around invariant closed subsets of paracompact spaces, as a consequence of equivariant-bundle classifying spaces being absolute neighborhood extensors in the relevant setting and (b) an equivariant-bundle version of Johnson's approximability of almost-multiplicative maps from finite-dimensional semisimple Banach algebras with Banach morphisms.