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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

C*-algebras associated with interval maps


Authors: Valentin Deaconu and Fred Shultz
Journal: Trans. Amer. Math. Soc. 359 (2007), 1889-1924
MSC (2000): Primary 46L80; Secondary 37E05
Published electronically: November 22, 2006
MathSciNet review: 2272154
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Abstract: For each piecewise monotonic map $ \tau$ of $ [0,1]$, we associate a pair of C*-algebras $ F_\tau$ and $ O_\tau$ and calculate their K-groups. The algebra $ F_\tau$ is an AI-algebra. We characterize when $ F_\tau$ and $ O_\tau$ are simple. In those cases, $ F_\tau$ has a unique trace, and $ O_\tau$ is purely infinite with a unique KMS state. In the case that $ \tau$ is Markov, these algebras include the Cuntz-Krieger algebras $ O_A$, and the associated AF-algebras $ F_A$. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and $ \beta$-transformations. For the case of interval exchange maps and of $ \beta$-transformations, the C*-algebra $ O_\tau$ coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.


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Additional Information

Valentin Deaconu
Affiliation: Department of Mathematics, University of Nevada, Reno, Nevada 89557
Email: vdeaconu@unr.edu

Fred Shultz
Affiliation: Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02481
Email: fshultz@wellesley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04112-2
PII: S 0002-9947(06)04112-2
Keywords: Dimension group, interval map, piecewise monotonic, unimodal map, tent map, Markov map, $\beta$-shift, interval exchange map, C*-algebra, Cuntz-Krieger algebra, Matsumoto algebra
Received by editor(s): August 1, 2004
Received by editor(s) in revised form: June 11, 2005
Published electronically: November 22, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.