## Illinois Journal of Mathematics

### 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.

#### Article information

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

Dates
First available in Project Euclid: 6 May 2014

https://projecteuclid.org/euclid.ijm/1399395828

Digital Object Identifier
doi:10.1215/ijm/1399395828

Mathematical Reviews number (MathSciNet)
MR3231479

Zentralblatt MATH identifier
1298.46053

#### Citation

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. https://projecteuclid.org/euclid.ijm/1399395828

#### References

• J.-C. Birget and J. Rhodes, Almost finite expansions of arbitrary semigroups, J. Pure Appl. Algebra 32 (1984), no. 3, 239–287.
• A. Buss and R. Exel, Twisted actions and regular Fell bundles over inverse semigroups, Proc. London Math. Soc. (3) 103 (2011), 235–270.
• R. Exel, Twisted partial actions: A classification of regular $C^*$-algebraic bundles, Proc. London Math. Soc. (3) 74 (1997), no. 2, 417–443.
• R. Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3481–3494.
• R. Exel, Noncommutative Cartan subalgebras of $C^*$-algebras, New York J. Math. 17 (2011), 331–382.
• J. Kellendonk and M. V. Lawson, Partial actions of groups, Internat. J. Algebra Comput. 14 (2004), no. 1, 87–114.
• M. V. Lawson, Inverse semigroups: The theory of partial symmetries, World Scientific, River Edge, NJ, 1998.
• M. V. Lawson, S. W. Margolis and B. Steinberg, Expansions of inverse semigroups, J. Aust. Math. Soc. 80 (2006), no. 2, 205–228.
• N. Sieben, $C^*$-crossed products by partial actions and actions of inverse semigroups, J. Austral. Math. Soc. Ser. A 63 (1997), no. 1, 32–46.
• N. Sieben, $C^*$-crossed products by twisted inverse semigroup actions, J. Operator Theory 39 (1998), no. 2, 361–393.
• M. B. Szendrei, A note on Birget–Rhodes expansion of groups, J. Pure Appl. Algebra 58 (1989), no. 1, 93–99.