KMS states on the $C^{\ast }$-algebras of reducible graphs. (11th August 2014)

• Record Type: Journal Article
• Authors: AN HUEF, ASTRID; LACA, MARCELO; RAEBURN, IAIN; SIMS, AIDAN
• Is Part Of: Ergodic theory and dynamical systems. Volume 35:Number 8(2015)
• Page Start: 2535
• Page End: 2558
• Publication Date: 2014-08-11
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2014.52
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : We consider the dynamics on the $C^{\ast }$ -algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on ${\mathcal{O}}_{A}$ . Math. Japon. 29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (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.