KMS states on C*-algebras associated to higher-rank graphs

摘要

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 T~k, and restricting these gauge actions to one-parameter subgroups of T~k 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 KMS β 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.