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In page 67 of Topology and Analysis by Booss and Bleecker, it is claimed that any Hilbert bundle is topologically trivial. Clearly, any smooth Hilbert bundle over a smooth manifold is topologically trivial, but it appears to be no reason to believe that this trivialization is smooth.My questions are:

  • Are there known conditions on the manifold or on the Hilbert space to guarantee that such topological trivialization is actually smooth?

  • Are there known counterexamples showing that such smooth trivialization is impossible in the general case?

EDIT: The definition of smooth Hilbert bundle is the one defined in 2.1 of: László Lempert and Róbert Szőke, Direct images, fields of Hilbert spaces, and geometric quantization, (arXiv:1004.4863) namely

A smooth (always complex) Hilbert bundle is a smooth map $p\colon H\to S$ of Banach manifolds, each fiber $p^{-1}(s)$, $s\in S$, is endowed with the structure of a complex vector space; for each $s\in S$ there should exist a neighborhood $U\subset S$, a complex Hilbert space $X$, and a smooth map (local trivialization) $F\colon p^{-1}U \to X$, whose restriction to each fiber $p^{-1}(t)$, $t\in U$, is linear, and such that $p\times F\colon p^{-1} U \to U \times X$ is a diffeomorphism.

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    $\begingroup$ What do you mean by a smooth Hilbert bundle? See eg mathoverflow.net/q/101526/4177 Note also that you need the structure group to be a Lie group, and $U(\mathcal{H})$ is a Banach Lie group in a norm topology, but not a Lie group in the strong topology (=compact-open topology). And the norm topology is in some sense "too strong". $\endgroup$ – David Roberts Oct 19 '17 at 23:52
  • $\begingroup$ @DavidRoberts I just edited the question to make explicit the definition of Hilbert bundle I'm using. It's the definition 2.1 of this paper: arxiv.org/pdf/1004.4863.pdf $\endgroup$ – Max Reinhold Jahnke Oct 20 '17 at 1:46
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I believe, the answer is (essentially) contained in the main theorem of the paper Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group by Christoph Müller and Christoph Wockel:

Let $K$ be a Lie group, modeled on a locally convex space, and $M$ a finite-dimensional paracompact manifold with corners. Then each continuous principal $K$-bundle over $M$ is equivalent to a smooth principal $K$-bundle. Moreover, two smooth principal $K$-bundles are continuously equivalent if and only if they are smoothly equivalent.

Now, every smooth Hilbert bundle in the sense of the post gives rise to a smooth $\mathrm{GL}(\mathcal H)$-principal bundle (defined, for instance, by the transition 1-cocycle) which is topologically trivial, hence by the above theorem smoothly trivial, and so therefore is the original vector bundle.

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