Equilibrium states on the Cuntz–Pimsner algebras of self-similar actions

Abstract

We consider a family of Cuntz-Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems.We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the Cuntz-Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group. 1 [26] I.

On Nica-Pimsner Algebras of C*-Dynamical Systems Over $\mathbb{Z}_+^n$

“…As in [24,26] we give a characterization of the ground states. This will imply their existence and association with the states on the C*-algebra A. Consequently, there is an affine homeomorphism from the state space S(A) onto the ground states on N T (A, α).…”

“…For the second part, we were intrigued by the ongoing program of Laca and Raeburn [24], as well as by the growing interest in the structure of KMS states on C*-algebras (deducting symmetry and/or phase transition breaking) from the seminal work of Bost and Connes [3]. See for example [1,8,13,15,21,22,23,24,25,26] to mention but a few inspiring works. For our analysis we follow the previous works of Laca and Neshveyev [22,23], Laca and Raeburn [24], and Laca, Raeburn, Ramagge and Whittaker [26].…”

“…See for example [1,8,13,15,21,22,23,24,25,26] to mention but a few inspiring works. For our analysis we follow the previous works of Laca and Neshveyev [22,23], Laca and Raeburn [24], and Laca, Raeburn, Ramagge and Whittaker [26]. Nevertheless we enrich our results by making use of the gauge invariant uniqueness theorems, and the automorphic dilation trick established in [9,Chapter 4].…”

KMS states on Pimsner algebras associated with C⁎-dynamical systems

“…The motivation for this work is two-fold. First of all we are inspired by the growing interest on the structure of the KMS states that involves: the Cuntz algebra [35], Cuntz-Krieger algebras [14], Hecke algebras [3], C*-algebras associated with subshifts [32], Pimsner algebras [26], the Toeplitz algebra of N ⋊ N × [27,28], C*-algebras of dilation matrices [29], C*-algebras of selfsimilar actions [30], topological dynamics [1,37,38], higher-rank graphs [2], and Nica-Pimsner algebras [21]. Secondly we are interested in analyzing further equivalence relations on multivariable classical systems [12,23].…”

“…Their approach is rather illuminating but it seems hard to be adopted in our specific example to provide a parametrization of the KMS states. It is the recent works of Laca and Raeburn [28], and Laca, Raeburn, Ramagge and Whittaker [30] that supply us with the efficient tools and algorithms for our purposes. We inform the reader that [26] follows Pimsner's setting [36], and the results about Cuntz-Pimsner algebras require a small reformulation to agree with the modern language of C*-correspondences set by Katsura [25].…”