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KMS states on the C∗-algebras of reducible graphs

Journal Article


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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 [KMS states for gauge action on OA. 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.

UOW Authors


  •   An Huef, Astrid (external author)
  •   Laca, Marcelo (external author)
  •   Raeburn, Iain F. (external author)
  •   Sims, Aidan

Publication Date


  • 2015

Citation


  • An Huef, A., Laca, M., Raeburn, I. & Sims, A. (2015). KMS states on the C∗-algebras of reducible graphs. Ergodic Theory and Dynamical Systems, 35 (8), 2535-2558.

Scopus Eid


  • 2-s2.0-84949316415

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=3791&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/2782

Number Of Pages


  • 23

Start Page


  • 2535

End Page


  • 2558

Volume


  • 35

Issue


  • 8

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 [KMS states for gauge action on OA. 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.

UOW Authors


  •   An Huef, Astrid (external author)
  •   Laca, Marcelo (external author)
  •   Raeburn, Iain F. (external author)
  •   Sims, Aidan

Publication Date


  • 2015

Citation


  • An Huef, A., Laca, M., Raeburn, I. & Sims, A. (2015). KMS states on the C∗-algebras of reducible graphs. Ergodic Theory and Dynamical Systems, 35 (8), 2535-2558.

Scopus Eid


  • 2-s2.0-84949316415

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=3791&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/2782

Number Of Pages


  • 23

Start Page


  • 2535

End Page


  • 2558

Volume


  • 35

Issue


  • 8