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Transactions of the American Mathematical Society
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
Published electronically: July 22, 2004
MathSciNet review: 2115083
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-04-03519-6
PII: S 0002-9947(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