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KMS states on the $C^{\ast }$-algebras of reducible graphs

Published online by Cambridge University Press:  11 August 2014

ASTRID AN HUEF ,
MARCELO LACA ,
IAIN RAEBURN  and
AIDAN SIMS
ASTRID AN HUEF
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email astrid@maths.otago.ac.nz, iraeburn@maths.otago.ac.nz
MARCELO LACA
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, Canada BC V8W 3P4 email laca@math.uvic.ca
IAIN RAEBURN
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email astrid@maths.otago.ac.nz, iraeburn@maths.otago.ac.nz
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email asims@uow.edu.au
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 8 , December 2015 , pp. 2535 - 2558
Copyright
© Cambridge University Press, 2014 

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