Equivariant Banach-bundle germs

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 U operating on all structures involved. We prove that in all cases homogeneous equivariant subbundles extend equivariantly from 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.