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

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 principle 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 principle 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 C⁎-algebra. This result gives nuclearity of a large class of C⁎-algebras as, remarkably, all the machinery in this paper works for étale non-Hausdorff groupoids just as well.