C*-algebras associated with interval maps

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_τare 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.