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

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- by R. Exel and J. Renault
- Proc. Amer. Math. Soc.
**134**(2006), 193-206 - DOI: https://doi.org/10.1090/S0002-9939-05-08129-3
- Published electronically: June 28, 2005
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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 $C^*(\mathcal {R})$.## References

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## Bibliographic 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