Real Analysis Exchange

KMS States, Entropy, and a Variational Principle for Pressure

Artur O. Lopes and Gilles G. de Castro

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We relate the concepts of entropy and pressure to that of KMS states for $C^*$-algebras. Several different definitions of entropy are known in our days: the one we present here is quite natural, extending the usual one for Dynamical Systems in Thermodynamic Formalism Theory, being basically obtained from transfer operators (also called Ruelle operators) and having the advantage of being very easily introduced. We also present a concept of pressure as a min-max principle. Later on, we consider the concept of a KMS state as an equilibrium state for a potential, in the context of $C^*$-algebras, and we show that there is a relation between equilibrium measures and KMS states for certain algebras arising from a continuous transformation.

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Real Anal. Exchange, Volume 34, Number 2 (2008), 333-346.

First available in Project Euclid: 29 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A35: Entropy and other invariants, isomorphism, classification 37A55: Relations with the theory of C-algebras [See mainly 46L55] 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 47B48: Operators on Banach algebras

entropy pressure KMS states transfer operator


de Castro, Gilles G.; Lopes, Artur O. KMS States, Entropy, and a Variational Principle for Pressure. Real Anal. Exchange 34 (2008), no. 2, 333--346.

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