Approximation properties of Fell bundles over inverse semigroups and non-Hausdorff groupoids

TL;DR

This paper investigates the nuclearity and weak containment of reduced cross-sectional C*-algebras of Fell bundles over inverse semigroups, extending key properties to non-Hausdorff groupoids and establishing new links between approximation properties and nuclearity.

Contribution

It develops an analogue of Fell's absorption trick for Fell bundles over inverse semigroups and proves that the approximation property implies nuclearity of the associated C*-algebras.

Findings

Fell's absorption trick extended to inverse semigroup bundles

Approximation property implies isomorphism of full and reduced C*-algebras

Nuclearity of cross-sectional C*-algebras under certain conditions

Abstract

In this paper we study the nuclearity and weak containment property of reduced cross-sectional C*-algebras of Fell bundles over inverse semigroups. In order to develop the theory, we first prove an analogue of Fell's absorption trick in the context of Fell bundles over inverse semigroups. In parallel, the approximation property of Exel can be reformulated in this context, and Fell's absorption trick can be used to prove that the approximation property, as defined here, implies that the full and reduced cross-sectional C*-algebras are isomorphic via the left regular representation, i.e., the Fell bundle has the weak containment property. We then use this machinery to prove that a Fell bundle with the approximation property and nuclear unit fiber has a nuclear cross-sectional \cstar{}algebra. This result gives nuclearity of a large class of C*-algebras, as, remarkably, all the machinery in this paper works for \'{e}tale non-Hausdorff groupoids just as well.