Some consequences of stabilization theorem for Fell bundles over exact groupoids

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 an appropriate sense. 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.