Illinois Journal of Mathematics

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

Alcides Buss and Ruy Exel

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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.

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Illinois J. Math., Volume 56, Number 4 (2012), 1185-1212.

First available in Project Euclid: 6 May 2014

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Zentralblatt MATH identifier

Primary: 20M18: Inverse semigroups 20M30: Representation of semigroups; actions of semigroups on sets 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]


Buss, Alcides; Exel, Ruy. Inverse semigroup expansions and their actions on $C^{\ast}$-algebras. Illinois J. Math. 56 (2012), no. 4, 1185--1212. doi:10.1215/ijm/1399395828.

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