KMS states on the operator algebras of reducible higher-rank graphs

TL;DR

This paper analyzes KMS states on the Toeplitz C*-algebra of reducible higher-rank graphs, revealing how their structure depends on graph components and spectral radii, with a complete analysis for certain temperature ranges.

Contribution

It provides a detailed analysis of KMS states for reducible higher-rank graphs using a specific dynamics, extending understanding beyond strongly connected cases.

Findings

KMS states depend on graph component relationships and spectral radii

Complete analysis of KMS states for inverse temperatures down to beta_c<1

Techniques applied to graphs with two connected components

Abstract

We study the equilibrium or KMS states of the Toeplitz C*-algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of the real numbers in the torus. Here we use an embedding which leads to a dynamics which has previously been identified as "preferred", and we scale the dynamics so that 1 is a critical inverse temperature. As with 1-graphs, we study the strongly connected components of the vertices of the graph. The behaviour of the KMS states depends on both the graphical relationships between the components and the relative size of the spectral radii of the vertex matrices of the components. We test our theorems on graphs with two connected components. We find that our techniques give a complete analysis of the KMS states with inverse temperatures down to a second critical temperature beta_c<1.