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 | References | Similar Articles | Additional Information

Abstract: Given a group , we construct, in a canonical way, an inverse semigroup associated to . The actions of are shown to be in one-to-one correspondence with the partial actions of , both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, and have the same representation theory. We show that governs the subsemigroup of all closed linear subspaces of a -graded -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 -algebra of a discrete group is introduced. While the usual group -algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group -algebra of the two commutative groups of order four, namely and , 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