KMS states on the -algebras of reducible graphs

Abstract

We consider the dynamics on the C * -algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single KMS state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.

KMS states on the Toeplitz algebras of higher-rank graphs

“…The structure of KMS-states for the gauge action or a generalised gauge action on the C * -algebra of a directed graph has revealed a complex structure, even for finite graphs [7,2]. The same is true for the Toeplitz C * -algebra of a finite directed graph [7,10], and it is therefore natural to seek for similar results for the canonical actions on the C * -algebra and the Toeplitz C *algebra of a finite higher-rank graph [12].…”

Equilibrium States on Graph Algebras

“…Very satisfactory results have been obtained for sytems associated to finite directed graphs, and we now have concrete descriptions of the simplices of KMS β states on the Toeplitz algebras at all inverse temperatures β [9,10]. Here we review these results, and discuss some surprising applications to work of Thomsen on systems involving the Cuntz-Pimsner algebras of local homeomeorphisms [17].…”

“…Since ρ(A E\H ) < ρ(A), we can also use Theorem 3.3 to find KMS ln ρ(A) states on T C * (E\H), and lift them to KMS ln ρ(A) states of T C * (E). Again the formulas and the complete classification are given in [10,Theorem 4.3].…”

“…One main conclusion of our recent work with Afsar [1] is that lower and upper bounds for the possible inverse temperatures given by Thomsen are sharp. For these appli-cations we do not need the full strength of the general results in [10], and in this article we describe a more direct approach.…”

KMS States on the Operator Algebras of Reducible Higher-Rank Graphs

“…∈ D} of j-coloured paths which make a quick exit from D. By Lemma 4.4 of [14], the projections {t λ t * λ : λ ∈ v QE j } are mutually orthogonal, and hence…”