Abstract
We study the generalised Bunce–Deddens algebras and their Toeplitz extensions constructed by Kribs and Solel from a directed graph and a sequence \(\omega \) of positive integers. We describe both of these \(C^*\)-algebras in terms of novel universal properties, and prove uniqueness theorems for them; if \(\omega \) determines an infinite supernatural number, then no aperiodicity hypothesis is needed in our uniqueness theorem for the generalised Bunce–Deddens algebra. We calculate the KMS states for the gauge action in the Toeplitz algebra when the underlying graph is finite. We deduce that the generalised Bunce–Deddens algebra is simple if and only if it supports exactly one KMS state, and this is equivalent to the terms in the sequence \(\omega \) all being coprime with the period of the underlying graph.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bartle, R.G.: The Elements of Integration. Wiley, New York (1966)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. Equilibrium states. Models in quantum statistical mechanics, vol. 2, p. xiv+519. Springer, Berlin (1997)
Bates, T., Pask, D., Raeburn, I., Szymański, W.: The \(C^*\)-algebras of row-finite graphs. N. Y. J. Math. 6, 307–324 (2000)
Choksi, J.R.: Inverse limits of measure spaces. Proc. Lond. Math. Soc. 8(3), 321–342 (1958)
Enomoto, M., Fujii, M., Watatani, Y.: KMS states for gauge action on \(O_{A}\). Math. Japon 29, 607–619 (1984)
Enomoto, M., Watatani, Y.: A graph theory for \(C^{\ast }\)-algebras. Math. Japon 25, 435–442 (1980)
Exel, R., Laca, M.: Partial dynamical systems and the KMS condition. Comm. Math. Phys. 232, 223–277 (2003)
Fowler, N.J., Raeburn, I.: The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48, 155–181 (1999)
an Huef, A., Raeburn, I.: The ideal structure of Cuntz-Krieger algebras. Ergod. Theory Dynam. Syst. 17, 611–624 (1997)
an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the \(C^*\)-algebras of finite graphs. J. Math. Anal. Appl. 405, 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, 265–283 (2014)
an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the \(C^*\)-algebras of reducible graphs. Ergod. Theory Dynam. Syst. 1–24 (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. 268, 1840–1875 (2015)
Katsura, T.: A class of \(C^*\)-algebras generalizing both graph algebras and homeomorphism \(C^*\)-algebras I, fundamental results. Trans. Am. Math. Soc. 356(11), 4287–4322 (2004)
Katsura, T.: A class of \(C^*\)-algebras generalizing both graph algebras and homeomorphism \(C^*\)-algebras II, examples. Int. J. Math. 17(7), 791–833 (2006)
Katsura, T.: A class of \(C^*\)-algebras generalizing both graph algebras and homeomorphism \(C^*\)-algebras III. Ideal structures. Ergod. Theory Dynam. Syst. 26, 1805–1854 (2006)
Kribs, D.W., Solel, B.: A class of limit algebras associated with directed graphs. J. Aust. Math. Soc. 82, 345–368 (2007)
Kumjian, A., Pask, D., Raeburn, I.: Cuntz-Krieger algebras of directed graphs. Pacific J. Math. 184, 161–174 (1998)
Laca, M., Larsen, N.S., Neshveyev, S., Sims, A., Webster, S.B.G.: Von Neumann algebras of strongly connected higher-rank graphs. Math. Ann. 363, 657–678 (2015)
Laca, M., Raeburn, I.: Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225, 643–688 (2010)
Li, H., Pask, D., Sims, A.: An elementary approach to \(C^*\)-algebras associated to topological graphs. N. Y. J. Math. 20, 447–469 (2014)
Raeburn, I.: Graph algebras. Published for the conference board of the mathematical sciences, p. vi+113. Washington, DC (2005)
Ruiz, E., Sims, A., Sørensen, A.P.W.: UCT-Kirchberg algebras have nuclear dimension 1. Adv. Math. 279, 1–28 (2015)
Ruiz, E., Sims, A., Tomforde, M.: The nuclear dimension of graph \(C^*\)-algebras. Adv. Math. 272, 96–123 (2015)
Seneta, E.: Non-negative matrices and Markov chains, Revised reprint of the second edtion (1981) (Springer, New York; MR0719544), p. xvi+287. Springer, New York (2006)
Acknowledgments
This research was supported by the Australian Research Council.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mohammad Sal Moslehian.
Rights and permissions
About this article
Cite this article
Robertson, D., Rout, J. & Sims, A. KMS States on Generalised Bunce–Deddens Algebras and their Toeplitz Extensions. Bull. Malays. Math. Sci. Soc. 41, 123–157 (2018). https://doi.org/10.1007/s40840-015-0244-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-015-0244-8