Equilibrium states and growth of quasi-lattice ordered monoids

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 βc such that at each inverse temperature β ≥ βc there exists a unique KMS state. Strictly above βc, 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 βc 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 −βc 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 −βc , 1), but we are not aware of any examples in which such roots exist.

𝐶*-algebras of right LCM monoids and their equilibrium states

“…Supplementing the study of semigroup C * -algebras from the point of view of amenability, nuclearity, and K-theory, there has been intense activity on describing KMS-states (in honour of Kubo and Martin-Schwinger) for certain natural flows on the C * -algebras, see [LR10,BaHLR12,CaHR16] as well as the very recent [ABLS17, BLRS19,ALN].…”

KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

“…In particular, if P is finitely generated and N (p) > 1 for all p in a finite generating set, then (8.6) is satisfied for all β large enough. Such β's can still be smaller than β c , see [4] and Example 9.6 below.…”

“…Consider the full semigroup C * -algebra of P , that is, the algebra N T (X) for the trivial product system X p = C, and the dynamics σ on it defined by N . Then, using our terminology, Theorem 3.5 and Proposition 3.5 in [4] can be formulated as follows:…”

“…This gives an answer to the second question in (2). Section 8 is inspired by the recent work of Bruce-Laca-Ramagge-Sims [4] on the Toeplitz algebras of quasi-lattice ordered monoids. Generalizing some of their results, we establish sufficient conditions for a KMS-state on N T (X) to be gauge-invariant, which can sometimes be applied to states at the critical inverse temperature and even below it.…”

“…Very recently, the analysis of KMS-states on the Toeplitz algebras of finite k-graphs has attained a very satisfactory form in Christensen's work [7] using the groupoid approach. Interesting results on KMS-states of the Toeplitz algebras of quasi-lattice ordered monoids, which can be considered as C * -algebras associated with the simplest rank one product systems, have been obtained in the work of Bruce-Laca-Ramagge-Sims [4]. This project started from the natural question whether it is possible to extend the theory developed in [21] to algebras associated with product systems.…”