Obstructed subhomogeneous-bundle extensions and embeddings

We address a number of problems concerning the (im)possibility of either extending locally trivial subbundles of possibly singular Banach/$C^*$ bundles globally, embedding subhomogeneous bundles into homogeneous ones, or recovering locally trivial compact-Lie-group-equivariant Banach or $C^*$ bundles as pullbacks along equivariant maps to compact spaces. The results include (1) the global extensibility of a locally trivial Banach/Hilbert/Banach-algebra/$C^*$ subbundle from a closed subspace of a paracompact space given appropriate homotopy constraints; (2) the homogeneous embeddability of equivariant subhomogeneous Banach/Hilbert bundles locally trivial along the singular locus under the same homotopy constraints, and (3) the characterization of finite-type equivariant locally trivial subhomogeneous $C^*$ bundles on normal spaces as precisely those (a) locally trivial as plain vector bundles, or (b) pulled back from the universal equivariant compactification or (c) pulled back from an equivariant map into a smooth manifold. The latter extends results of Phillips concerning non-equivariant matrix-algebra bundles restricted along the Stone\v{C}ech compactification.