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

 

$AF$-algebras and the tail-equivalence relation on Bratteli diagrams


Authors: R. Exel and J. Renault
Journal: Proc. Amer. Math. Soc. 134 (2006), 193-206
MSC (2000): Primary 46L05, 46L85
Published electronically: June 28, 2005
MathSciNet review: 2170559
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Abstract: We show that the $C^*$-algebra associated to the tail-equivalence relation on a Bratteli diagram, according to a procedure recently introduced by the first-named author and A. Lopes, is isomorphic to the $AF$-algebra of the diagram. More generally we consider an approximately proper equivalence relation $\mathcal{R}=\bigcup_{n\in\mathbb{N} }\mathcal{R}_n$ on a compact space $X$ for which the quotient maps $\pi_n\colon X\to X/\mathcal R_n$ are local homeomorphisms. We show that the algebra associated to $\mathcal{R}$ under the above-mentioned procedure is isomorphic to the groupoid $C^*$-algebra $C^*(\mathcal{R})$.


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Additional Information

R. Exel
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88010-970 Florianópolis, Brasil
Email: exel@mtm.ufsc.br

J. Renault
Affiliation: Département de Mathématiques, Université d’Orléans, France
Email: renault@labomath.univ-orleans.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08129-3
PII: S 0002-9939(05)08129-3
Received by editor(s): April 26, 2004
Received by editor(s) in revised form: August 24, 2004
Published electronically: June 28, 2005
Additional Notes: The first author was partially supported by CNPq
The second author was partially supported by Réseau pour les mathématiques, Coopération Franco-Brésilienne
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.