Limits of KMS states on Toeplitz algebras of finite graphs

TL;DR

This paper investigates the structure of KMS states on Toeplitz algebras of finite graphs, clarifying the roles of non-minimal components and the relationships between vertices and minimal components at critical inverse temperatures.

Contribution

It extends previous work by analyzing the influence of non-minimal components and the connections between vertices and minimal components in the KMS-structure.

Findings

KMS states at critical inverse temperatures decompose into convex combinations of states from minimal components.

The coefficient of a KMS state associated to a minimal component is nonzero iff a maximal path exists from that component to the vertex.

The structure of extremal KMS states of type I corresponds to vertices, while those at critical temperatures relate to minimal strongly connected components.

Abstract

The structure of KMS states of Toeplitz algebras associated to finite graphs equipped with the gauge action is determined by an Huef--Laca--Raeburn--Sims. Their results imply that extremal KMS states of type I correspond to vertices, while extremal KMS states at critical inverse temperatures correspond to minimal strongly connected components. The purpose of this article is to clarify the role of non-minimal components and the relation between vertices and minimal components in the KMS-structure. For each component $C_0$ and each vertex $v \in C_0$, the KMS states at the critical inverse temperature of $v$ obtained by the limit of type I KMS states associated to $v$ uniquely decomposes into a convex combination of KMS states associated to minimal components. We show that for each minimal component $C$, the coefficient of the KMS state associated to $C$ is nonzero if and only if there exists a maximal path from $C$ to $C_0$ in the graph of components.