  We investigate recent uniqueness theorems for reduced $C^*$-algebras of
Hausdorff \'{e}tale groupoids in the context of inverse semigroups. In many
cases the distinguished subalgebra is closely related to the structure of the
inverse semigroup. In order to apply our results to full $C^*$-algebras, we
also investigate amenability. More specifically, we obtain conditions that
guarantee amenability of the universal groupoid for certain classes of inverse
semigroups. These conditions also imply the existence of a conditional
expectation onto a canonical subalgebra.
