A non-locally trivial ${\mathrm{W}}^{\ast }$-bundle with fixed factorial fibres

In this paper we construct the first example of a non-locally trivial ${\mathrm{W}}^{\ast }$-bundle whose fibres are all isomorphic to some fixed ${\mathrm{I}\mathrm{I}}_1$ factor. This is achieved by introducing a notion of uniformly having spectral gap for ${\mathrm{W}}^{\ast }$-bundles. For bundles with fixed factorial fibres, the negation of having this uniform spectral gap property provides an obstruction for being locally trivial. This results in seemingly elementary examples of ${\mathrm{W}}^{\ast }$-bundles whose fibres are all isomorphic to some fixed factor but that are not locally trivial, even over spaces with covering dimension equal to zero.