Associativity of crossed products by partial actions, enveloping actions and partial representations

Dokuchaev, M.; Exel, R.

doi:10.1090/S0002-9947-04-03519-6

Associativity of crossed products by partial actions, enveloping actions and partial representations
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by M. Dokuchaev and R. Exel PDF

Trans. Amer. Math. Soc. 357 (2005), 1931-1952

Abstract:

Given a partial action $\alpha$ of a group $G$ on an associative algebra $\mathcal {A}$, we consider the crossed product $\mathcal {A}\rtimes _\alpha G$. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of $\mathcal {A}\rtimes _\alpha G$ obtained in the context of $C^*$-algebras. In particular, we prove that $\mathcal {A} \rtimes _{\alpha } G$ is associative, provided that $\mathcal {A}$ is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

References

F. Abadie, Sobre ações parcias, fibrados de Fell e grupóides, PhD Thesis, Universidade de São Paulo, 1999.
Fernando Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), no. 1, 14–67. MR 1957674, DOI 10.1016/S0022-1236(02)00032-0
Yu. A. Bahturin, S. K. Sehgal, and M. V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2001), no. 2, 677–698. MR 1843319, DOI 10.1006/jabr.2000.8643
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Michael Dokuchaev, Ruy Exel, and Paolo Piccione, Partial representations and partial group algebras, J. Algebra 226 (2000), no. 1, 505–532. MR 1749902, DOI 10.1006/jabr.1999.8204
M. Dokuchaev, N. Zhukavets, On finite degree partial representations of groups, J. Algebra, 274 (2004), 309–334.
M. Dokuchaev, N. Zhukavets, On irreducible partial representations of groups, Comptes Rendus Math. Rep. Acad. Sci. Canada, 24 (2002), 85–90.
Ruy Exel, Circle actions on $C^*$-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence, J. Funct. Anal. 122 (1994), no. 2, 361–401. MR 1276163, DOI 10.1006/jfan.1994.1073
Ruy Exel, Twisted partial actions: a classification of regular $C^*$-algebraic bundles, Proc. London Math. Soc. (3) 74 (1997), no. 2, 417–443. MR 1425329, DOI 10.1112/S0024611597000154
Ruy Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3481–3494. MR 1469405, DOI 10.1090/S0002-9939-98-04575-4
Ruy Exel, Amenability for Fell bundles, J. Reine Angew. Math. 492 (1997), 41–73. MR 1488064, DOI 10.1515/crll.1997.492.41
Ruy Exel and Marcelo Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119–172. MR 1703078, DOI 10.1515/crll.1999.051
J. M. G. Fell and R. S. Doran, Representations of $^*$-algebras, locally compact groups, and Banach $^*$-algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125, Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras. MR 936628
Peter A. Fillmore, A user’s guide to operator algebras, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1996. A Wiley-Interscience Publication. MR 1385461
Kevin McClanahan, $K$-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), no. 1, 77–117. MR 1331978, DOI 10.1006/jfan.1995.1064
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
John Quigg and Iain Raeburn, Characterisations of crossed products by partial actions, J. Operator Theory 37 (1997), no. 2, 311–340. MR 1452280
Louis H. Rowen, Ring theory, Student edition, Academic Press, Inc., Boston, MA, 1991. MR 1095047
S. K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. MR 1242557