Non-commutative branched covers and bundle unitarizability

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 posed by Blanchard–Gogić (in the metrizable case); (b) in general, these expectations cannot have an “optimal index,” thereby giving a negative answer to 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 . The latter result quantitatively resolves another problem posed by Gogić.