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ISSN 1088-6850(e) ISSN 0002-9947(p)

C*-algebras associated with interval maps

Author(s): Valentin Deaconu; Fred Shultz
Journal: Trans. Amer. Math. Soc. 359 (2007), 1889-1924.
MSC (2000): Primary 46L80; Secondary 37E05
Posted: November 22, 2006
MathSciNet review: 2272154
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Abstract: For each piecewise monotonic map $\tau$ of , 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 , and the associated AF-algebras . 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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