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 Szymański. We study the KMS equilibrium states of C ^*-dynamical systems based on 𝒪 _A and 𝒯 _A , with dynamics satisfying for the canonical generating partial isometries s _x and arbitrary real numbers N _x > 1. The KMS_β states on both 𝒪 _A and 𝒯 _A are completely characterized for certain values of the inverse temperature β, according to the position of β relative 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.