  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.
