Inverse semigroup expansions and their actions on C^-algebras

In this work, we give a presentation of the prefix expansion Pr (G) of an inverse semigroup G as recently introduced by Lawson, Margolis and Steinberg which is similar to the universal inverse semigroup defined by the second named author in case G is a group. The inverse semigroup Pr (G) classifies the partial actions of G on spaces. We extend this result and prove that Fell bundles over G correspond bijectively to saturated Fell bundles over Pr (G). In particular, this shows that twisted partial actions of G (on C*-algebras) correspond to twisted (global) actions of Pr (G). Furthermore, we show that this correspondence preserves C*-algebra crossed products.