von Neuman algebras of strongly connected higher-rank graphs

TL;DR

This paper analyzes the types of extremal KMS states for strongly connected finite higher-rank graphs, revealing a temperature-dependent dichotomy in their algebraic structure, including type I and type III factors.

Contribution

It characterizes the factor types of extremal KMS states for higher-rank graph algebras, especially at critical inverse temperature, linking them to graph spectral properties.

Findings

All extremal KMS states are type I∞ for inverse temperatures above 1.

At inverse temperature 1, the KMS state type depends on the graph structure, being either type I or type III.

The Connes invariant for type III factors is computed using spectral radii and cycle degrees.

Abstract

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 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 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.