Noncommutative Pressure and the Variational Principle in Cuntz–Krieger-type C*-Algebras

Abstract

Let a be a self-adjoint element of an exact C g -algebra A and h: A Q A a contractive completely positive map. We define a notion of dynamical pressure P h (a) which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu-Brown topological entropy and Neshveyev and Størmer unital-nuclear pressure. A variational inequality bounding P h (a) below by the free energies h s (h)+s(a) with respect to the Sauvageot-Thouvenot entropy h s (h) is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C g -algebras furnish a framework for investigating the variational principle, which asserts the equality of P h (a) with the supremum of the free energies over all h-invariant states. In one direction we extend Brown's result on the constancy of the Voiculescu-Brown entropy upon passing to the crossed product, and in another we show that the pressure of a self-adjoint element over the Markov subshift underlying the canonical map on the Cuntz-Krieger algebra O A is equal to its classical pressure. The latter result is extended to a more general setting comprising an expanded class of Cuntz-Krieger-type Pimsner algebras, leading to the variational principle for self-adjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz-Krieger algebras.

KMS states on C*-algebras associated to expansive maps

“…Our results, which complete [R4, Section 4.2], strictly cover previous results about KMS states of gauge one-parameter automorphism groups of Cuntz and Cuntz-Krieger algebras obtained by a number of authors (e.g. [OP,Ev,EFW,Z,KP,Ex2]). Although they do not use groupoid techniques as we do, Kerr-Pinzari in [KP] and Exel in [Ex2] also make essential use of Ruelle's transfer operator.…”

“…KMS states with respect to gauge automorphism groups on CuntzKrieger algebras have also been studied by D. Kerr and C. Pinzari in [KP,Section 7] and by R. Exel in [Ex2]. Given a primitive n×n matrix A with zero-one elements, we define X = X A and T = T A as in Example 2.3 and…”

Entropy for automorphisms of free groups

“…If β is an automorphism of an exact C * -algebra, then ht(β) ≥ h φ (β) by [8,Proposition 9], and also for the modified dynamical entropy ht φ (β) of ht(β), we have that ht φ (β) ≤ ht(β) by [5] (cf. [9]). Furthermore, if φ is a tracial state, then h φ (β) = H(β) by [7].…”

Lyapunov spectra for KMS states on Cuntz-Krieger algebras

“…The novelty here is that we are able to include the case in which the underlying dynamical system is expansive, and hence our approach generalises the well-studied purely hyperbolic (expanding) situation to systems with parabolic elements. More precisely, recall that in [20] and [25] it was shown that if A is irreducible and H > 0 then there is a one-one correspondence between the set of (H, β)-KMS states and the set of eigenmeasures of L * −βH associated with the eigenvalue 1. We add to this by showing that even for H ≥ 0 we still have that there exists a bijection from the set of eigenmeasures to the set of (H, β)-KMS states which are trivial on {H = 0} (see Fact 8,Fact 9).…”

“…However, if there are parabolic elements then additional to the Patterson-KMS state at δ(G) there exist further (βJ, 1)-KMS states for each β > δ(G) which all correspond to purely atomic measures concentrated on the fixed points of the parabolic transformations (see [38], [40]; also, for related investigations in terms of C * -algebras for expanding dynamical systems see e.g. [30], [16], [33], [25], [17], [27], [20], [26]). In contrast to this usual approach, in this paper we additionally allow additive pertubations of the potential functions.…”