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 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 of A such that for all pairs a,b∈ the C*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ such that [αj(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∈, 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.