Locally Trivial W*-Bundles

Abstract

We prove that a tracially continuous W$^*$-bundle $\mathcal{M}$ over a compact Hausdorff space $X$ with all fibres isomorphic to the hyperfinite II$_1$-factor $\mathcal{R}$ that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group $\mathrm{Aut}({\mathcal{R}})$ shown by Popa and Takesaki. There is no restriction on the covering dimension of $X$.