von Neuman algebras of strongly connected higher-rank graphs

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 \(k\)-graph. For inverse temperatures above 1, all of the extremal KMS states are of type \(\mathrm {I}_\infty \). At inverse temperature 1, there is a dichotomy: if the \(k\)-graph is a simple \(k\)-dimensional cycle, we obtain a finite type \(\mathrm {I}\) 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.