The Variational Principle for a Class of Asymptotically Abelian C*-Algebras |
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Authors: | Sergey Neshveyev Erling Størmer |
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Affiliation: | (1) Institute for Low Temperature Physics and Engineering, 47, Lenin Ave., 310164 Kharkov, Ukraine, UA;(2) Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway, NO |
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Abstract: | Let (A,α) be a C*-dynamical system. We introduce the notion of pressure P α(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 ? of A such that for all pairs a,b∈? the C*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α j (a),b]= 0 for |j|≥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 α with respect to the α-invariant state φ. If H∈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. Received: 19 April 2000 / Accepted: 14 June 2000 |
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