  We study simplicity and pure infiniteness criteria for C*-algebras associated
to inverse semigroup actions by Hilbert bimodules and to Fell bundles over
etale not necessarily Hausdorff groupoids. Inspired by recent work of Exel and
Pitts, we introduce essential crossed products for which there are such
criteria. In our approach the major role is played by a generalised expectation
with values in the local multiplier algebra. We give a long list of equivalent
conditions characterising when the essential and reduced C*-algebras coincide.
Our most general simplicity and pure infiniteness criteria apply to aperiodic
C*-inclusions equipped with supportive generalised expectations. We thoroughly
discuss the relationship between aperiodicity, detection of ideals, purely
outer inverse semigroup actions, and non-triviality conditions for dual
groupoids.
