Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-05-08129-3
Published electronically: June 28, 2005
MathSciNet review: 2170559
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $C^*(\mathcal{R})$.


References [Enhancements On Off] (What's this?)

  • 1. O. Bratteli, Inductive limits of finite dimensional $C^*$-algebras, Trans. Amer. Math. Soc., 171 (1972), 195-234. MR 0312282 (47:844)
  • 2. R. Exel and A. Lopes, $C^*$-algebras, approximately proper equivalence relations, and thermodynamics formalism, Ergodic Theory Dynam. Systems, 24 (2004), 1051-1082. MR 2085390
  • 3. P. S. Muhly, J. Renault, and D. P. Williams, Equivalence and isomorphism for groupoid $C^*$-algebras, J. Operator Theory, 17 (1987), 3-22. MR 0873460 (88h:46123)
  • 4. P. S. Muhly and D. P. Williams, Continuous trace groupoid $C^*$-algebras, Math. Scand., 66 (1990), 231-241. MR 1075140 (91j:46081)
  • 5. J. Renault, A groupoid approach to $C^*$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, 1980. MR 0584266 (82h:46075)
  • 6. J. Renault, The Radon-Nikodym problem for approximately proper equivalence relations, Ergodic Theory Dynam. Systems, to appear, [arXiv:math.OA/0211369].
  • 7. Y. Watatani, Index for $C^*$-subalgebras, Mem. Am. Math. Soc., 424 (1990), 117 pp. MR 0996807 (90i:46104)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L05, 46L85

Retrieve articles in all journals with MSC (2000): 46L05, 46L85


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: https://doi.org/10.1090/S0002-9939-05-08129-3
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.

American Mathematical Society