Serre-Swan theorem for Hilbert bundles?

Question

Is there a version of Serre-Swan theorem for infinite rank Hilbert bundles? For example, let $X$ be a smooth compact manifold and $E\to X$, $F\to X$ are infinite rank Hilbert bundles. If such a theorem existed, it would imply, among other things, $E\cong F$ if and only if $\Gamma(X, E)\cong\Gamma(X, F)$ as $C(X)$-modules.

Answer 1

There are different aspects here. To prove that $E\cong F$ if and only if $\Gamma(X,E)\cong\Gamma(X,F)$ as $C(X)$ modules seems to be much easier than a full version of the Serre-Swan theorem. As far as I can see, the only fact you need in order to prove the first statement is that a map $\Gamma(X,E)\to\Gamma(X,F)$ which is linear over $C(X)$ is induced by a bundle map between the bundles. It seems to me that the standard proof that if $\Phi$ is such that map then $\Phi(s(x))$ depends only on $s(x)$ extends to the setting of infinite rank bundles without problems. You'd probably have to add an appropriate continuity condition the setting of sections to ensure continuity of the induced bundle map, but this looks rather harmless to me.

To obtain a full version of the Serre-Swann Theorem, you would need an algebraic characterization of the modules $\Gamma(X,E)$ (which replaces standard condition "finitely generated projective"). This seems much harder to me unless you work in some restricted class of Hilbert bundles (say involving a $C^*$-algebra).