Von Neumann algebras of strongly connected higher-rank graphs

Abstract

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.

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Correspondence to Aidan Sims.

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This research was supported by the Australian Research Council and the Natural Sciences and Engineering Research Council of Canada. Parts of the work were completed at the workshop Operator algebras and dynamical systems from number theory (13w5152) at the Banff International Research Station in November 2013 and at the conference Classification, Structure, Amenability and Regularity at the University of Glasgow in September 2014, supported by the EPSRC, GMJT and LMS.

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Laca, M., Larsen, N.S., Neshveyev, S. et al. Von Neumann algebras of strongly connected higher-rank graphs. Math. Ann. 363, 657–678 (2015). https://doi.org/10.1007/s00208-015-1187-y

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Mathematics Subject Classification

  • Primary 46L10
  • Secondary 46L05