KMS states on Nica-Toeplitz C*-algebras

Abstract

Given a quasi-lattice ordered group $(G,P)$ and a compactly aligned product system $X$ of essential C${}^{\ast }$-correspondences over the monoid $P$, we show that there is a bijection between the gauge-invariant KMS${}_{\beta }$-states on the Nica-Toeplitz algebra $\mathcal{N}\mathcal{T}(X)$ of $X$ with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra $A$ satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on $P$ and $X$, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $\beta$. Under fairly general additional assumptions we show that there is a critical inverse temperature ${\beta }_c$ such that for $\beta >{\beta }_c$ all KMS${}_{\beta }$-states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of KMS${}_{\beta }$-states in terms of tracial states on $A$, while at $\beta ={\beta }_c$ we have a phase transition manifesting itself in the appearance of KMS${}_{\beta }$-states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on $A$ can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $\mathcal{N}\mathcal{T}(X)$.

arXiv comment45 pages; v2: minor fixes, a few improvements (in particular, a shorter proof of the KMS condition and an example showing that the small system of inequalities for free abelian monoids is optimal)

References in corpus (7)

• KMS states of quasi-free dynamics on Pimsner algebras
• KMS states on the C*-algebras of non-principal groupoids
• KMS States, Entropy and the Variational Principle in full C*-dynamical systems
• On Bost-Connes type systems for number fields
• Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems
• Equilibrium states and growth of quasi-lattice ordered monoids
• Symmetries of the KMS simplex

Cited by in corpus (3)

• The groupoid approach to equilibrium states on right LCM semigroup C*-algebras
• Equilibrium on Toeplitz extensions of higher dimensional noncommutative tori
• KMS spectra for group actions on compact spaces