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Equilibrium states on higher-rank Toeplitz non-commutative solenoids

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  17 April 2019

ZAHRA AFSAR Open the ORCID record for ZAHRA AFSAR [Opens in a new window] ,
ASTRID AN HUEF Open the ORCID record for ASTRID AN HUEF [Opens in a new window] ,
IAIN RAEBURN Open the ORCID record for IAIN RAEBURN [Opens in a new window]  and
AIDAN SIMS Open the ORCID record for AIDAN SIMS [Opens in a new window]
ZAHRA AFSAR
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia email [email protected]
ASTRID AN HUEF
Affiliation:
School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington6140, New Zealand email [email protected], [email protected]
IAIN RAEBURN
Affiliation:
School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington6140, New Zealand email [email protected], [email protected]
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email [email protected]
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Abstract

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.

Keywords

KMS statesC∗ -algebrasdirect limitToeplitz algebra

MSC classification

Primary: 46L05: General theory of $C^*$-algebras 46L30: States 46L55: Noncommutative dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 2881 - 2912
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Copyright
© Cambridge University Press, 2019

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References

Afsar, Z., Brownlowe, N., Larsen, N. S. and Stammeier, N.. Equilibrium states on right LCM semigroup C ∗ -algebras. Int. Math. Res. Not. 2019(6) (2019), 1642–1698.CrossRefGoogle Scholar
Afsar, Z., an Huef, A. and Raeburn, I.. KMS states on C ∗ -algebras associated to local homeomorphisms. Internat. J. Math. 25 (2014), article no. 1450066, 28 pp.CrossRefGoogle Scholar
Afsar, Z., an Huef, A. and Raeburn, I.. KMS states on C ∗ -algebras associated to a family of ∗-commuting local homeomorphisms. J. Math. Anal. Appl. 464 (2018), 965–1009.CrossRefGoogle Scholar
Afsar, Z., Larsen, N. S. and Neshveyev, S.. KMS states on Nica–Toeplitz algebras. Preprint, 2018, arXiv:1807.05822.Google Scholar
Baggett, L. W., Larsen, N. S., Packer, J. A., Raeburn, I. and Ramsay, A.. Direct limits, multiresolution analyses, and wavelets. J. Funct. Anal. 258 (2010), 2714–2738.CrossRefGoogle Scholar
Brownlowe, N., Hawkins, M. and Sims, A.. The Toeplitz noncommutative solenoid and its Kubo–Martin–Schwinger states. Ergod. Th. & Dynam. Sys. 39 (2019), 105–131.CrossRefGoogle Scholar
Christensen, J.. Symmetries of the KMS simplex. Comm. Math. Phys. 364 (2018), 357–383.CrossRefGoogle Scholar
Christensen, J.. KMS states on the Toeplitz algebras of higher-rank graphs. Preprint, 2018, arXiv:1805.09010.Google Scholar
Cuntz, J., Deninger, C. and Laca, M.. C ∗ -algebras of Toeplitz type associated with algebraic number fields. Math. Ann. 355 (2013), 1383–1423.CrossRefGoogle Scholar
Davidson, K. R.. C ∗ -Algebras by Example (Fields Institute Monographs, 6) . American Mathematical Society, Providence, RI, 1996.CrossRefGoogle Scholar
Exel, R. and Laca, M.. Partial dynamical systems and the KMS condition. Comm. Math. Phys. 232 (2003), 223–277.CrossRefGoogle Scholar
Fell, J. M. G. and Doran, R. S.. Representations of ∗-Algebras. Locally Compact Groups, and Banach ∗-Algebraic Bundles, Vol. 1. Academic Press, San Diego, CA, 1988.Google Scholar
Fletcher, J., an Huef, A. and Raeburn, I.. A program for finding all KMS states on the Toeplitz algebra of a higher-rank graph. J. Operator Theory, to appear, Preprint, 2018, arXiv:1801.03189.Google Scholar
Folland, G. B.. Real Analysis: Modern Techniques and their Applications, 2nd edn. Wiley, New York, 1999.Google Scholar
an Huef, A., Laca, M., Raeburn, I. and Sims, A.. KMS states on the C ∗ -algebras of finite graphs. J. Math. Anal. Appl. 405 (2013), 388–399.CrossRefGoogle Scholar
an Huef, A., Laca, M., Raeburn, I. and Sims, A.. KMS states on C ∗ -algebras associated to higher-rank graphs. J. Funct. Anal. 266 (2014), 265–283.CrossRefGoogle Scholar
Kakariadis, E. T. A.. On Nica–Pimsner algebras of C ∗ -dynamical systems over ℤ+ n . Int. Math. Res. Not. IMRN 2017(4) (2017), 1013–1065.Google Scholar
Kakariadis, E. T. A.. Equilibrium states and entropy theory for Nica–Pimsner algebras. Preprint, 2018, arXiv:1806.02443.Google Scholar
Laca, M. and Neshveyev, S.. Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers. J. Funct. Anal. 261 (2011), 169–187.CrossRefGoogle Scholar
Laca, M. and Raeburn, I.. Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139 (1996), 415–440.CrossRefGoogle Scholar
Laca, M. and Raeburn, I.. Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225 (2010), 643–688.CrossRefGoogle Scholar
Laca, M., Raeburn, I., Ramagge, J. and Whittaker, M. F.. Equilibrium states on the Cuntz–Pimsner algebras of self-similar actions. J. Funct. Anal. 266 (2014), 6619–6661.CrossRefGoogle Scholar
Laca, M., Raeburn, I., Ramagge, J. and Whittaker, M. F.. Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs. Adv. Math. 331 (2018), 268–325.CrossRefGoogle Scholar
Latrémolière, F. and Packer, J. A.. Noncommutative solenoids. New York J. Math. 24a (2018), 155–191.Google Scholar
Nica, A.. C ∗ -algebras generated by isometries and Wiener–Hopf operators. J. Operator Theory 27 (1992), 17–52.Google Scholar
Raeburn, I.. On crossed products by coactions and their representation theory. Proc. Lond. Math. Soc. (3) 64 (1992), 625–652.CrossRefGoogle Scholar