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Morita equivalence of partial group actions and globalization


Authors: F. Abadie, M. Dokuchaev, R. Exel and J. J. Simón
Journal: Trans. Amer. Math. Soc. 368 (2016), 4957-4992
MSC (2010): Primary 16S35; Secondary 16W22, 46L05
DOI: https://doi.org/10.1090/tran/6525
Published electronically: November 6, 2015
MathSciNet review: 3456167
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Abstract: We consider a large class of partial actions of groups on rings, called regular, which contains all $ s$-unital partial actions as well as all partial actions on $ C^{\ast }$-algebras. For them the notion of Morita equivalence is introduced, and it is shown that any regular partial action is Morita equivalent to a globalizable one and that the globalization is essentially unique. It is also proved that Morita equivalent $ s$-unital partial actions on rings with orthogonal local units are stably isomorphic. In addition, we show that Morita equivalent $ s$-unital partial actions on commutative rings must be isomorphic, and an analogous result for $ C^{\ast }$-algebras is also established.


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Additional Information

F. Abadie
Affiliation: Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225, CP 11400, Montevideo, Uruguay
Email: fabadie@cmat.edu.uy

M. Dokuchaev
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, 05508-090 São Paulo, SP, Brasil
Email: dokucha@ime.usp.br

R. Exel
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brasil
Email: exel@mtm.ufsc.br

J. J. Simón
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30071 Murcia, España
Email: jsimon@um.es

DOI: https://doi.org/10.1090/tran/6525
Keywords: Partial action, skew group ring, Morita equivalence, $C^{\ast}$-algebra
Received by editor(s): June 5, 2013
Received by editor(s) in revised form: June 2, 2014
Published electronically: November 6, 2015
Additional Notes: This work was partially supported by CNPq of Brazil (Proc. 305975/2013-7, Proc. 300362/2010-2), Fapesp of Brazil (Proc. Proc. 2009/52665-0), MINECO (Ministerio de Economía y Competitividad), (Fondo Europeo de Desarrollo Regional) project MTM2012-35240, Spain, and Fundación Séneca of Murcia, Programa Hispano Brasileño de Cooperación Universitaria PHB2012-0135, Spain
Article copyright: © Copyright 2015 American Mathematical Society