KMS states on [formula omitted]-algebras associated to higher-rank graphs

Abstract

Consider a higher-rank graph of rank k. Both the Cuntz–Krieger algebra and the Toeplitz–Cuntz–Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these gauge actions to one-parameter subgroups of Tk gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMSβ states on the Toeplitz–Cuntz–Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature βc: for β larger than βc, there is a d-dimensional simplex of KMS states; when β=βc and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz–Krieger algebra. As in previous studies for k=1, our main tool is the Perron–Frobenius theory for irreducible non-negative matrices, though here we need a version of the theory for commuting families of matrices.