Amenability for actions of étale groupoids on $C^{\ast }$-algebras and Fell bundles

Abstract

We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable $C^{\ast }$-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the $C^{\ast }$-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel's approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid actions, we show that the approximation property implies measurewise amenability of the underlying action.

33 pages. Updated references and added remarks about the non-étale case. Accepted for publication in Trans. Amer. Math. Soc