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Equilibrium states and growth of quasi-lattice ordered monoids


Authors: Chris Bruce, Marcelo Laca, Jacqui Ramagge and Aidan Sims
Journal: Proc. Amer. Math. Soc. 147 (2019), 2389-2404
MSC (2010): Primary 46L10; Secondary 46L05
DOI: https://doi.org/10.1090/proc/14108
Published electronically: March 1, 2019
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Abstract: Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz $ C^*$-algebra; here we study the KMS equilibrium states of the resulting $ C^*$-dynamical system. We show that under a nondegeneracy assumption on the homomorphism there is a critical inverse temperature $ \beta _c$ such that at each inverse temperature $ \beta \geq \beta _c$ there exists a unique KMS state. Strictly above $ \beta _c$, the KMS states are generalised Gibbs states with density operators determined by analytic extension to the upper half-plane of the unitaries implementing the dynamics. These are faithful Type I states. The critical value $ \beta _c$ is the largest real pole of the partition function of the system and is related to the clique polynomial and skew-growth function of the monoid, relative to the degree map given by the logarithm of the multiplicative homomorphism. Motivated by the study of equilibrium states, we give a proof of the inversion formula for the growth series of a quasi-lattice ordered monoid in terms of the clique polynomial as in recent work of Albenque-Nadeau and McMullen for the finitely generated case and in terms of the skew-growth series as in recent work of Saito. Specifically, we show that $ e^{-\beta _c}$ is the smallest pole of the growth series and thus is the smallest positive real root of the clique polynomial. We use this to show that equilibrium states in the subcritical range can only occur at inverse temperatures that correspond to roots of the clique polynomial in the interval $ (e^{-\beta _c},1)$, but we are not aware of any examples in which such roots exist.


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Additional Information

Chris Bruce
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Email: cmbruce@uvic.ca

Marcelo Laca
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Email: laca@uvic.ca

Jacqui Ramagge
Affiliation: School of Mathematics and Statistics, Sydney University, New South Wales 2006, Australia
Email: jacqui.rammage@sydney.edu.au

Aidan Sims
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia
Email: asims@uow.edu.au

DOI: https://doi.org/10.1090/proc/14108
Keywords: KMS states, quasi-lattice ordered group, Artin monoid
Received by editor(s): October 30, 2017
Received by editor(s) in revised form: January 26, 2018
Published electronically: March 1, 2019
Additional Notes: This research was supported by the Natural Sciences and Engineering Research Council of Canada and Australian Research Council grants DP150101595 and DP170101821.
Part of this work was completed while the second and fourth authors were attending the MATRIX@Melbourne Research Program “Refining $C^{*}$-algebraic invariants for dynamics using $KK$-theory”, July 18–29 2016.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2019 American Mathematical Society