Amenability for actions of \'etale groupoids on $C^*$-algebras and Fell bundles

TL;DR

This paper extends the concept of measurewise amenability to actions of étale groupoids on $C^*$-algebras and Fell bundles, establishing conditions under which nuclearity of crossed products and cross-sectional algebras is characterized.

Contribution

It introduces a generalized notion of measurewise amenability for groupoid actions and Fell bundles, linking it to nuclearity and the approximation property in these contexts.

Findings

Measurewise amenability characterizes nuclearity of crossed products.

The approximation property implies nuclearity of the cross-sectional algebra.

The approximation property ensures measurewise amenability for groupoid actions.

Abstract

We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, \'etale groupoids on separable $C^*$-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the $C^*$-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, \'etale 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.