We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz-Krieger algebra of a strongly connected finite (Formula presented.)-graph. For inverse temperatures above 1, all of the extremal KMS states are of type (Formula presented.). At inverse temperature 1, there is a dichotomy: if the (Formula presented.)-graph is a simple (Formula presented.)-dimensional cycle, we obtain a finite type (Formula presented.) factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph.
Additional Grant Number
Laca, M., Larsen, N. S., Neshveyev, S., Sims, A. D. & Webster, S. B. (2015). Von Neumann algebras of strongly connected higher-rank graphs. Mathematische Annalen, 363 (1), 657-678.