Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

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


Authors: M. Dokuchaev and R. Exel
Journal: Trans. Amer. Math. Soc. 357 (2005), 1931-1952
MSC (2000): Primary 16S99; Secondary 16S10, 16S34, 16S35, 16W22, 16W50, 20C07, 20L05
DOI: https://doi.org/10.1090/S0002-9947-04-03519-6
Published electronically: July 22, 2004
MathSciNet review: 2115083
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. F. Abadie, Sobre ações parcias, fibrados de Fell e grupóides, PhD Thesis, Universidade de São Paulo, 1999.
  • 2. F. Abadie, Enveloping Actions and Takai Duality for Partial Actions, J. Funct. Anal. 197 (2003), 14-67. MR 2004c:46130
  • 3. Yu. Bahturin, S. K. Sehgal, M. V. Zaicev, Group gradings on associative algebras, J. Algebra, 241 (2001), 677-698. MR 2002h:16067
  • 4. J. Cuntz, W. Krieger, A Class of $C^*$-Algebras and Topological Markov Chains, Inventiones Math. 56 (1980), 251-268. MR 82f:46073a
  • 5. M. Dokuchaev, R. Exel, P. Piccione, Partial representations and partial group algebras, J. Algebra, 226 (2000), 505-532. MR 2001m:16034
  • 6. M. Dokuchaev, N. Zhukavets, On finite degree partial representations of groups, J. Algebra, 274 (2004), 309-334.
  • 7. M. Dokuchaev, N. Zhukavets, On irreducible partial representations of groups, Comptes Rendus Math. Rep. Acad. Sci. Canada, 24 (2002), 85-90.
  • 8. R. Exel, Circle actions on $C^*$-algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequences, J. Funct. Anal. 122 (1994), 361-401. MR 95g:46122
  • 9. R. Exel, Twisted partial actions: a classification of regular $C^*$-algebraic bundles, Proc. London Math. Soc. 74 (1997), 417-443. MR 98d:46075
  • 10. R. Exel, Partial Actions of Groups and Actions of Semigroups, Proc. Am. Math. Soc. 126 (1998), 3481-3494. MR 99b:46102
  • 11. R. Exel, Amenability for Fell Bundles, J. Reine Angew. Math. 492 (1997), 41-73. MR 99a:46131
  • 12. R. Exel, M. Laca, Cuntz-Krieger Algebras for Infinite Matrices, J. Reine Angew. Math. 512 (1999), 119-172. MR 2000i:46064
  • 13. J. M. G. Fell, P. S. Doran, Representations of *-Algebras, Locally Compact Groups and Banach *-Algebraic Bundles I, Academic Press, 1988, Pure and Applied Math. 125. MR 90c:46001
  • 14. P. A. Fillmore, A User's Guide to Operator Algebras, Willey - Interscience, 1996. MR 97i:46094
  • 15. K. McClanahan, K-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), 77-117. MR 96i:46083
  • 16. D. S. Passman, The Algebraic Structure of Group Rings, Interscience, New York, 1977. MR 81d:16001
  • 17. J. C. Quigg, I. Raeburn, Characterizations of Crossed Products by Partial Actions, J. Operator Theory 37 (1997), 311-340. MR 99a:46121
  • 18. L. H. Rowen, Ring theory - Student edition, Academic Press, 1991. MR 94e:16001
  • 19. S. K. Sehgal, Units in Integral Group Rings, Longman Scientific & Technical Press, Harlow, 1993. MR 94m:16039

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16S99, 16S10, 16S34, 16S35, 16W22, 16W50, 20C07, 20L05

Retrieve articles in all journals with MSC (2000): 16S99, 16S10, 16S34, 16S35, 16W22, 16W50, 20C07, 20L05


Additional Information

M. Dokuchaev
Affiliation: Departamento de Matemática, Universidade de São Paulo, Brazil
Email: dokucha@ime.usp.br

R. Exel
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, Brazil
Email: exel@mtm.ufsc.br

DOI: https://doi.org/10.1090/S0002-9947-04-03519-6
Keywords: Partial action, crossed product, partial representation, partial group ring, grading, groupoid
Received by editor(s): February 19, 2003
Received by editor(s) in revised form: September 26, 2003
Published electronically: July 22, 2004
Additional Notes: This work was partially supported by CNPq of Brazil (Proc. 301115/95-8, Proc. 303968/85-0)
Article copyright: © Copyright 2004 M. Dokuchaev and R. Exel

American Mathematical Society