Morita enveloping Fell bundles

TL;DR

This paper introduces equivalence notions for Fell bundles over groups, demonstrating that all Fell bundles are equivalent to semidirect product bundles, which helps extend results on crossed products and amenability.

Contribution

It defines weak and strong equivalence for Fell bundles and proves their preservation of key properties, extending known results to a broader class of bundles.

Findings

Every Fell bundle is strongly or weakly equivalent to a semidirect product bundle.

Equivalences preserve cross-sectional C*-algebras and amenability.

Results on crossed products and amenability extend to Fell bundles.

Abstract

We introduce notions of weak and strong equivalence for non-saturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional $C^*-$algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.