The variational principle for a class of asymptotically abelian C*-algebras

Abstract

Let (A,α) be a C*-dynamical system. We introduce the notion of pressure P_α(H) of the automorphism αat a self-adjoint operator H\in A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra \A of A such that for all pairs a,b\in\A the C*-algebra they generate is finite dimensional, and there is p=p(a,b)\in\N such that [α^j(a),b]=0 for |j|\ge p. For systems in this class we prove the variational principle, i.e. show that P_α(H) is the supremum of the quantities h_ϕ(α)-ϕ(H), where h_ϕ(α) is the Connes-Narnhofer-Thirring dynamical entropy of αwith respect to the α-invariant state ϕ. If H\in\A, and P_α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h_ϕ(α), and any state of finite maximal entropy is a trace.