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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
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

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
Copyright
© Cambridge University Press, 2014 

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