Mathematische Annalen

, Volume 363, Issue 1–2, pp 657–678 | Cite as

Von Neumann algebras of strongly connected higher-rank graphs

  • Marcelo Laca
  • Nadia S. Larsen
  • Sergey Neshveyev
  • Aidan Sims
  • Samuel B. G. Webster


We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz–Krieger algebra of a strongly connected finite \(k\)-graph. For inverse temperatures above 1, all of the extremal KMS states are of type \(\mathrm {I}_\infty \). At inverse temperature 1, there is a dichotomy: if the \(k\)-graph is a simple \(k\)-dimensional cycle, we obtain a finite type \(\mathrm {I}\) factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph.

Mathematics Subject Classification

Primary 46L10 Secondary 46L05 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Marcelo Laca
    • 1
  • Nadia S. Larsen
    • 2
  • Sergey Neshveyev
    • 2
  • Aidan Sims
    • 3
  • Samuel B. G. Webster
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia

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