Inverse semigroup expansions and their actions on $C^{\ast}$-algebras

Abstract

In this work, we give a presentation of the prefix expansion ${\operatorname{\mathbf {Pr}} (G)}$ 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 ${\operatorname{\mathbf {Pr}} (G)}$ 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 ${\operatorname{\mathbf {Pr}} (G)}$. In particular, this shows that twisted partial actions of $G$ (on $C^{*}$-algebras) correspond to twisted (global) actions of ${\operatorname{\mathbf {Pr}} (G)}$. Furthermore, we show that this correspondence preserves $C^{*}$-algebra crossed products.