C*-algebras associated with interval maps

Abstract

For each piecewise monotonic map τ \tau of [ 0 , 1 ] [0,1] , we associate a pair of C*-algebras F τ F_\tau and O τ O_\tau and calculate their K-groups. The algebra F τ F_\tau is an AI-algebra. We characterize when F τ F_\tau and O τ O_\tau are simple. In those cases, F τ F_\tau has a unique trace, and O τ O_\tau is purely infinite with a unique KMS state. In the case that τ \tau is Markov, these algebras include the Cuntz-Krieger algebras O A O_A , and the associated AF-algebras F A F_A . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and β \beta -transformations. For the case of interval exchange maps and of β \beta -transformations, the C*-algebra O τ O_\tau coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.