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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References
Bratteli, O.: Inductive limits of finite dimensional \(C^{\ast }\)-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)
Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. In: Equilibrium States. Models in Quantum Statistical Mechanics, vol. 2. Springer, Berlin (1997). (xiv+519)
Carlsen, T.M., Kang, S., Shotwell, J., Sims, A.: The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources. J. Funct. Anal. 266(4), 2570–2589 (2014)
Connes, A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. (4) 6, 133–252 (1973)
Davidson, K.R., Yang, D.: Periodicity in rank 2 graph algebras. Can. J. Math. 61(6), 1239–1261 (2009)
Enomoto, M., Fujii, M., Watatani, Y.: KMS states for gauge action on \({\cal O_{A}}\). Math. Jpn. 29(4), 607–619 (1984)
Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Am. Math. Soc. 234(2), 289–324 (1977)
Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras. II. Trans. Am. Math. Soc. 234(2), 325–359 (1977)
Gao, N., Troitsky, V.G.: Irreducible semigroups of positive operators on Banach lattices. Linear Multilinear Algebra 62(1), 74–95 (2014)
Hazlewood, R., Raeburn, I., Sims, A., Webster, S.B.G.: Remarks on some fundamental results about higher-rank graphs and their \(C^{*}\)-algebras. Proc. Edinb. Math. Soc. (2) 56(2), 575–597 (2013)
an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the \(C^{*}\)-algebras of finite graphs. J. Math. Anal. Appl. 405(2), 388–399 (2013)
an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on C\(^{*}\)-algebras associated to higher-rank graphs. J. Funct. Anal. 266(1), 265–283 (2014)
an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the C\(^{*}\)-algebra of a higher-rank graph and periodicity in the path space. J. Funct. Anal. (2015). (To appear)
Kitchens, B.P.: Symbolic dynamics, One-sided, Two-sided and Countable State Markov Shifts. Springer, Berlin (1998). (x+252)
Kumjian, A., Pask, D.: Higher rank graph \(C^{*}\)-algebras. N. Y. J. Math. 6, 1–20 (2000)
Kumjian, A., Pask, D.: Actions of \(\mathbb{Z}^k\) associated to higher rank graphs. Ergod. Theory Dyn. Syst. 23(4), 1153–1172 (2003)
Maloney, B., Pask, D.: Simplicity of the \(C^{*}\)-algebras of skew product \(k\)-graphs. (2013). arXiv:1306.6107 [math.OA]
Neshveyev, S.: KMS states on the \(C^{*}\)-algebras of non-principal groupoids. J. Oper. Theory 70(2), 513–530 (2013)
Neshveyev, S., Størmer, E.: Dynamical entropy in operator algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. In: A Series of Modern Surveys in Mathematics. Springer, Berlin (2006)
Okayasu, R.: Type III factors arising from Cuntz–Krieger algebras. Proc. Am. Math. Soc. 131(7), 2145–2153 (2003)
Olesen, D., Pedersen, G.K.: Some \(C^{\ast } \)-dynamical systems with a single KMS state. Math. Scand. 42(1), 111–118 (1978)
Raeburn, I., Sims, A.: Product systems of graphs and the Toeplitz algebras of higher-rank graphs. J. Oper. Theory 53(2), 399–429 (2005)
Seneta, E.: Non-negative matrices and Markov chains. In: Springer Series in Statistics, 2nd edn. Springer, New York (1981)
Yang, D.: Type III von Neumann algebras associated with 2-graphs. Bull. Lond. Math. Soc. 44(4), 675–686 (2012)
Yang, D.: Factoriality and type classification of \(k\)-graph von Neumann algebras. (2013). arXiv:1311.4638 [math.OA]
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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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DOI: https://doi.org/10.1007/s00208-015-1187-y