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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Partial crossed product description of the $ C^*$-algebras associated with integral domains


Authors: Giuliano Boava and Ruy Exel
Journal: Proc. Amer. Math. Soc. 141 (2013), 2439-2451
MSC (2010): Primary 46L05, 46L55
Published electronically: April 3, 2013
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Abstract: Recently, Cuntz and Li introduced the $ C^*$-algebra $ \mathfrak{A}[R]$ associated to an integral domain $ R$ with finite quotients. In this paper, we show that $ \mathfrak{A}[R]$ is a partial group algebra of the group $ K\rtimes K^\times $ with suitable relations, where $ K$ is the field of fractions of $ R$. We identify the spectrum of these relations and we show that it is homeomorphic to the profinite completion of $ R$. By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li. Among them, we prove that $ \mathfrak{A}[R]$ is simple by showing that the action is topologically free and minimal.


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Additional Information

Giuliano Boava
Affiliation: Instituto Nacional de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro/RJ, Brazil
Address at time of publication: Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis/SC, Brazil
Email: gboava@gmail.com

Ruy Exel
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis/SC, Brazil
Email: r@exel.com.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11724-7
PII: S 0002-9939(2013)11724-7
Received by editor(s): May 23, 2011
Received by editor(s) in revised form: October 22, 2011
Published electronically: April 3, 2013
Additional Notes: The first author’s research was supported by CNPq, Brazil
The second author’s research was partially supported by CNPq, Brazil
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.