KMS states on the C*-algebra of a higher-rank graph and periodicity in the path space

TL;DR

This paper characterizes KMS states on the C*-algebra of a strongly connected finite k-graph, linking their existence to the graph's periodicity and simplicity, and revealing phase transitions in Toeplitz extensions.

Contribution

It provides a complete classification of KMS states for these algebras, connecting state existence to graph properties and periodicity, and addresses a question posed by Yang.

Findings

Unique KMS state when the algebra is simple.

KMS states parameterized by characters of an abelian group.

Phase change phenomena in Toeplitz extensions when algebra is not simple.

Abstract

We study the KMS states of the C*-algebra of a strongly connected finite k-graph. We find that there is only one 1-parameter subgroup of the gauge action that can admit a KMS state. The extreme KMS states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite-path space of the graph. We deduce that there is a unique KMS state if and only if the k-graph C*-algebra is simple, giving a complete answer to a question of Yang. When the k-graph C*-algebra is not simple, our results reveal a phase change of an unexpected nature in its Toeplitz extension.