C*-algebras associated with interval maps

Deaconu, Valentin; Shultz, Fred

doi:10.1090/S0002-9947-06-04112-2

C*-algebras associated with interval maps
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Trans. Amer. Math. Soc. 359 (2007), 1889-1924 Request permission

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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