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

Full-text PDF Free Access

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.

**[1]**B. Abadie, S. Eilers and R. Exel, ``Morita equivalence and crossed products by Hilbert -bimodules'', preprint, Universidade de São Paulo, 1994, to appear in Trans. Amer. Math. Soc. CMP**97:17****[2]**B. A. Barnes, Representations of the algebra of an inverse semigroup, Trans. Amer. Math. Soc.**218**, (1976), 361-396. MR**53:1169****[3]**J. Duncan and A. L. T. Paterson, ``-algebras of inverse semigroups'', Proc. Edinburgh Math. Soc.**28**(1985), 41-58. MR**86h:46090****[4]**R. Exel, ``Twisted Partial Actions, A Classification of Regular -Algebraic Bundles'', Proc. London Math. Soc.**74**(1997), 417-443. CMP**97:05****[5]**R. Exel, ``Circle Actions on -Algebras, Partial Automorphisms and a Generalized Pimsner-Voiculescu Exact Sequence'', J. Funct. Analysis**122**(1994), 361-401. MR**95g:46122****[6]**J. M. G. Fell and R. S. Doran, ``Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles'', Pure and Applied Mathematics. vol. 125 and 126, Academic Press, 1988. MR**90c:46001**; MR**90c:46002****[7]**J. M. Howie, ``An introduction to Semigroup theory'', Academic Press, 1976. MR**57:6235****[8]**K. Jensen and K. Thomsen, ``Elements of -Theory'', Birkhäuser, 1991. MR**94b:19008****[9]**K. McClanahan, ``-theory for partial crossed products by discrete groups'', J. Funct. Analysis,**130**(1995), 77-117. MR**95i:46083****[10]**G. K. Pedersen, ``-Algebras and their automorphism groups'', Acad. Press, 1979. MR**81e:46037****[11]**M. Pimsner, ``A class of -algebras generalizing both Cuntz-Krieger algebras and crossed products by**Z**'', Amer. Math. Soc., Providence, R.I., 1997. MR**97k:46069****[12]**J. Quigg and I. Raeburn, ``Landstad duality for partial actions'', preprint, University of Newcastle, 1994.**[13]**J. Renault, ``A groupoid approach to -algebras'', Lecture Notes in Mathematics, vol. 793, Springer, 1980. MR**82h:46075****[14]**N. Sieben, ``-crossed products by partial actions and actions of inverse semigroups'', Masters Thesis, Arizona State University, 1994. CMP**97:14**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
20M18,
46L05,
20M30

Retrieve articles in all journals with MSC (1991): 20M18, 46L05, 20M30

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