Partial Dynamical Systems and the KMS Condition

Abstract

Given a countably infinite 0-1 matrix A without identically zero rows, let O_A be the Cuntz-Krieger algebra recently introduced by the authors and T_A be the Toeplitz extension of O_A, once the latter is seen as a Cuntz-Pimsner algebra, as recently shown by Szymanski. We study the KMS equilibrium states of C*-dynamical systems based on O_A and T_A, with dynamics satisfying σ_t(s_x) = N_x^{it} s_x for the canonical generating partial isometries s_x and arbitrary real numbers N_x > 1. The KMS_βstates on both O_A and T_A are completely characterized for certain values of the inverse temperature β, according to the position of βrelative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for O_A are derived from those for T_A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for O_n and of Enomoto, Fujii and Watatani for O_A as particular cases.