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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Partial actions of groups and actions of inverse semigroups
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by Ruy Exel PDF
Proc. Amer. Math. Soc. 126 (1998), 3481-3494 Request permission


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
  • MR Author ID: 239607
  • Email:
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3481-3494
  • MSC (1991): Primary 20M18, 46L05, 20M30
  • DOI:
  • MathSciNet review: 1469405