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

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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.