Some consequences of the stabilization theorem for Fell bundles over exact groupoids

TL;DR

This paper explores the implications of a stabilization theorem for Fell bundles over exact groupoids, establishing conditions for nuclearity and exactness of associated $C^*$-algebras and characterizing exact groupoids.

Contribution

It demonstrates that groupoid exactness is equivalent to Fell exactness and extends the understanding of exactness preservation under extensions of groupoids.

Findings

Conditions for nuclearity and exactness of Fell bundle $C^*$-algebras

Equivalence of groupoid exactness and Fell exactness

Extensions of exact groupoids are also exact

Abstract

We investigate some consequences of a recent stabilization result of Ionescu, Kumjian, Sims, and Williams, which says that every Fell bundle $C^*$-algebra is Morita equivalent to a canonical groupoid crossed product. First we use the theorem to give conditions that guarantee the $C^*$-algebras associated to a Fell bundle are either nuclear or exact. We then show that a groupoid is exact if and only if it is "Fell exact", in the sense that any invariant ideal gives rise to a short exact sequence of reduced Fell bundle $C^*$-algebras. As an application, we show that extensions of exact groupoids are exact by adapting a recent iterated Fell bundle construction due to Buss and Meyer.