KMS states on the Toeplitz algebras of higher-rank graphs

Abstract

The Toeplitz algebra T C * (Λ) for a finite k-graph Λ is equipped with a continuous one-parameter group α r for each r ∈ R k , obtained by composing the map R ∋ t → (e itr 1 , . . . , e itr k ) ∈ T k with the gauge action on T C * (Λ). In this paper we give a complete description of the β-KMS states for the C * -dynamical system (T C * (Λ), α r ) for all finite k-graphs Λ and all values of β ∈ R and r ∈ R k .1. Λ has no sinks and no sources.

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

“…For finite k-graphs part (i) of the above proposition recovers [7,Proposition 4.3] that was proved using the groupoid picture. Together with the theory developed in Section 10, leading to Example 10.6, this provides an alternative route to the classification of gauge-invariant KMSstates in [7,Theorem 5.9]. Part (ii) recovers [15, Theorem 6.1(a)].…”

“…Throughout this section we fix a quasi-lattice ordered group (G, P ) of the form (G 1 , P 1 )×· · ·× (G n , P n ), a compactly aligned product system X of C * -correspondences over P , with X e = A, a homomorphism N : P → (0, +∞), and β ∈ R. In this setting, developing the ideas in [17,7], we can refine the decomposition of states on N T (X) into positive functionals of finite and infinite types by considering states that are finite with respect to some factors and infinite with respect to other.…”

“…C * -algebras associated to k-graphs are another large class of examples closely related to product systems whose structure of KMS-states has been extensively studied in the recent years starting with [15]. 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].…”

“…His approach is quite different from ours and relies on a more refined decomposition of KMS-states than just into finite/infinite types. Namely, a key idea in [17], partly inspired by [7], is that a state can be of finite type with respect to one coordinate of Z n + and of infinite type with respect to another. We explain how this idea works for product-monoids in our approach and as an application show how results in [17] and our results for Z n + can be quickly deduced from each other, see Example 10.6.…”

Equilibrium states on higher-rank Toeplitz non-commutative solenoids

“…Typically, these Toeplitz extensions exhibit more interesting KMS structure. This recent work has covered Toeplitz algebras of directed graphs and their higherrank analogues [15,16,7,13,8] (after earlier work in [11]), Toeplitz algebras arising in number theory [9], the Nica-Toeplitz extensions of Cuntz-Pimsner algebras [19,17,18,1,4], and Toeplitz algebras associated to self-similar actions [22,23]. In [6], Brownlowe, Hawkins and Sims described Toeplitz extensions of the noncommutative solenoids from [24], and considered a natural dynamics on this extension.…”