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

Let (A,α) be a C∗-dynamical system. We introduce the notion of pressure Pα(H) of the automorphism α at a self-adjoint operator H ∈ 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 ∈ A the C∗-algebra they generate is finite dimensional, and there is p = p(a, b) ∈ N such that [α(a), b] = 0 for |j| ≥ p. For systems in this class we prove the variational principle, i.e. show that Pα(H) is the supremum of the quantities hφ(α)−φ(H), where hφ(α) is the Connes-Narnhofer-Thirring dynamical entropy of α with respect to the α-invariant state φ. If H ∈ A, and Pα(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 hφ(α), and any state of finite maximal entropy is a trace.