2025/11/17 by Chirvasitu, Alexandru
#46L85 #46M20 #54C55 #54D20 #55M15 #55R10 #55R15 #55R91 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2511.13511
Consider a continuous bundle E→ X of Banach/Hilbert spaces or Banach/C^*-algebras over a paracompact base space, equivariant for a compact Lie group \mathbbU operating on all structures involved. We prove that in all cases homogeneous equivariant subbundles extend equivariantly from \mathbbU-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.