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

Abstract

AbstractGiven a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$ C ∗ -correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$ KMS β -states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$ NT ( 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 $$\beta $$ β . Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$ β c such that for $$\beta >\beta _c$$ β > β c all $$\hbox {KMS}_\beta $$ KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$ KMS β -states in terms of tracial states on A, while at $$\beta =\beta _c$$ β = β c we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$ KMS β -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 $$\mathcal {NT}(X)$$ NT ( X ) .