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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$AF$-algebras and the tail-equivalence relation on Bratteli diagrams
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by R. Exel and J. Renault PDF
Proc. Amer. Math. Soc. 134 (2006), 193-206 Request permission

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
  • MR Author ID: 239607
  • Email: exel@mtm.ufsc.br
  • J. Renault
  • Affiliation: Département de Mathématiques, Université d’Orléans, France
  • MR Author ID: 146950
  • Email: renault@labomath.univ-orleans.fr
  • 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
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 193-206
  • MSC (2000): Primary 46L05, 46L85
  • DOI: https://doi.org/10.1090/S0002-9939-05-08129-3
  • MathSciNet review: 2170559