Locally Trivial W*-Bundles

TL;DR

This paper proves that a locally trivial W*-bundle with hyperfinite II_1-fiber over a compact space must be globally trivial, leveraging the contractibility of automorphism groups without dimension restrictions.

Contribution

It establishes the global triviality of certain W*-bundles over compact spaces, extending previous results by removing dimension constraints.

Findings

Locally trivial W*-bundles with hyperfinite II_1-fibers are globally trivial.

Uses contractibility of automorphism group of hyperfinite II_1-factor.

No restriction on the covering dimension of the base space.

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 $X$.