Equivalence of Fell bundles over groups

TL;DR

This paper introduces a new notion of equivalence for Fell bundles over groups, demonstrating that equivalent bundles have Morita-Rieffel equivalent cross-sectional $C^*$-algebras and that this equivalence preserves amenability.

Contribution

It defines a novel equivalence concept for Fell bundles, extending previous notions, and proves its properties including Morita-Rieffel equivalence and preservation of amenability.

Findings

Equivalent Fell bundles have Morita-Rieffel equivalent cross-sectional $C^*$-algebras.

Amenability is preserved under the new equivalence.

Equivalence of Fell bundles is an equivalence relation.

Abstract

We give a notion of equivalence for Fell bundles over groups, not necessarily saturated nor separable, and show that equivalent Fell bundles have Morita-Rieffel equivalent cross-sectional $C^*$-algebras. Our notion is originated in the context of partial actions and their enveloping actions. The equivalence between two Fell bundles is implemented by a bundle of Hilbert bimodules with some extra structure. Suitable cross-sectional spaces of such a bundle turn out to be imprimitivity bimodules for the cross-sectional $C^*$-algebras of the involved Fell bundles. We show that amenability is preserved under this equivalence and, by means of a convenient notion of internal tensor product between Fell bundles, we show that equivalence of Fell bundles is an equivalence relation.