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Journal of Operator Theory

Volume 81, Issue 2, Spring 2019  pp. 335-369.

Some consequences of stabilization theorem for Fell bundles over exact groupoids

Authors:  Scott M. LaLonde
Author institution:Department of Mathematics, The University of Texas at Tyler, Tyler, TX 75799, U.S.A.

Summary: 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.

Keywords: Fell bundle, exact groupoid, groupoid crossed product, nuclearity, exactness

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