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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Partial actions of groups and actions
of inverse semigroups


Author: Ruy Exel
Journal: Proc. Amer. Math. Soc. 126 (1998), 3481-3494
MSC (1991): Primary 20M18, 46L05, 20M30
DOI: https://doi.org/10.1090/S0002-9939-98-04575-4
MathSciNet review: 1469405
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Abstract: Given a group $G$, we construct, in a canonical way, an inverse semigroup $\mathcal{S}(G)$ associated to $G$. The actions of $\mathcal{S}(G)$ are shown to be in one-to-one correspondence with the partial actions of $G$, both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, $G$ and $\mathcal{S}(G)$ have the same representation theory. We show that $\mathcal S(G)$ governs the subsemigroup of all closed linear subspaces of a $G$-graded ${C}^*$-algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A ``partial'' version of the group ${ C}^*$-algebra of a discrete group is introduced. While the usual group ${ C}^*$-algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group ${ C}^*$-algebra of the two commutative groups of order four, namely $Z/4 Z$ and $ Z/2 Z \oplus Z/2 Z$, are not isomorphic.


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Additional Information

Ruy Exel
Affiliation: Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, 05508-900 São Paulo, Brazil
Address at time of publication: Departamento de Matemática, Universidade Federal de Santa Catarina, 88010-970 Florianópolis, SC, Brazil
Email: exel@mtm.ufsc.br

DOI: https://doi.org/10.1090/S0002-9939-98-04575-4
Received by editor(s): June 19, 1996
Received by editor(s) in revised form: April 16, 1997
Additional Notes: The author was partially supported by CNPq, Brazil.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society