Abstract
Given a topologically free action of a countable group G on a compact metric space X, there is a canonical correspondence between continuous 1-cocycles for this group action and diagonal 1-parameter groups of automorphisms of the reduced crossed product C\(^*\)-algebra. The KMS spectrum is defined as the set of inverse temperatures for which there exists a KMS state. We prove that the possible KMS spectra depend heavily on the nature of the acting group G. For groups of subexponential growth, we prove that the only possible KMS spectra are \(\{0\}\), \([0,+\infty )\), \((-\infty ,0]\) and \({\mathbb {R}}\). For certain wreath product groups, which are amenable and of exponential growth, we prove that any closed subset of \({\mathbb {R}}\) containing zero arises as KMS spectrum. Finally, for certain nonamenable groups including the free group with infinitely many generators, we prove that any closed subset may arise. Besides uncovering a surprising relation between geometric group theoretic properties and KMS spectra, our results provide two simple C\(^*\)-algebras with the following universality property: any closed subset (containing, resp. not containing zero) arises as the KMS spectrum of a 1-parameter group of automorphisms of this C\(^*\)-algebra.
Similar content being viewed by others
References
an Huef, A., Laca, M., Raeburn, I., Sims, A.: KMS states on the C\(^{*}\)-algebras of reducible graphs. Ergod. Theory Dyn. Syst 35, 2535–2558 (2015)
Afsar, Z., Larsen, N., Neshveyev, S.: KMS states on Nica–Toeplitz C\(^{*}\)-algebras. Commun. Math. Phys. 378, 1875–1929 (2020)
Boivin, D., Derriennic, Y.: The ergodic theorem for additive cocycles of \(\mathbb{Z}^d\) or \(\mathbb{R}^d\). Ergod. Theory Dyn. Syst. 11, 19–39 (1991)
Bratteli, O., Elliott, G.A., Herman, R.H.: On the possible temperatures of a dynamical system. Commun. Math. Phys. 74, 281–295 (1980)
Bratteli, O., Elliott, G., Kishimoto, A.: The temperature state space of a dynamical system I. J. Yokohama Univ. 28, 125–167 (1980)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I, II. Texts and Monographs in Physics, Springer, Berlin (1981)
Christensen, J., Thomsen, K.: KMS states on the crossed product C\(^*\)-algebra of a homeomorphism. Ergod. Theory Dyn. Syst. (to appear). arXiv:1912.06069
Denker, M., Urbanski, M.: On the existence of conformal measures. Trans. Am. Math. Soc. 328, 563–587 (1991)
Kumjian, A., Renault, J.: KMS states on \(C^{*}\)- algebras associated to expansive maps. Proc. Am. Math. Soc. 134, 2067–2078 (2006)
Neshveyev, S.: KMS states on the C\(^*\)-algebras of non-principal groupoids. J. Operator Theory 70, 513–530 (2011)
Olesen, D., Pedersen, G.K.: Some C\(^{*}\)-dynamical systems with a single KMS state. Math. Scand. 42, 111–118 (1978)
Renault, J.: A Groupoid Approach to C\(^*\)-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)
Thomsen, K.: KMS weights on graph \(C^*\)-algebras. Adv. Math. 309, 334–391 (2017)
Thomsen, K.: Phase transition in the CAR algebra. Adv. Math. 372, 27 (2020)
Thomsen, K.: On the possible temperatures for flows on a UHF algebra. arXiv:2012.03306
Acknowledgements
The proof of Theorem A uses unpublished ideas developed by Klaus Thomsen and the first named author while working on [CT19], which handles the special case \(G={\mathbb {Z}}\). We are grateful to Klaus Thomsen for allowing us to include them in this article, and for discussions leading to the results in Sect. 5.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Ogata.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Johannes Christensen Supported by a DFF-International Postdoctoral Grant. Stefaan Vaes Supported by FWO research project G090420N of the Research Foundation Flanders and by long term structural funding—Methusalem Grant of the Flemish Government.
Rights and permissions
About this article
Cite this article
Christensen, J., Vaes, S. KMS Spectra for Group Actions on Compact Spaces. Commun. Math. Phys. 390, 1341–1367 (2022). https://doi.org/10.1007/s00220-021-04282-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-021-04282-w