C∗-Algebras and Thermodynamic Formalism

We show a relation of the KMS state of a certain C∗Algebra U with the Gibbs state of Thermodynamic Formalism. More precisely, we consider here the shift T : X → X acting on the Bernoulli space X = {1, 2, ..., k} and μ a Gibbs state defined by a Holder continuous potential p : X → R, and L(μ) the associated Hilbert space. Consider the C∗-Algebra U = U(μ), which is a sub-C∗-Algebra of the C∗-Algebra of linear operators in L(μ) which will be precisely defined later. We call μ the reference measure. Consider a fixed Holder potential H > 0 and the C-dynamical system defined by the associated homomorphism σt. We are interested in describe for such system the KMS states φβ for all β ∈ R. We show a relation of a new Gibbs probability νβ to a KMS state φνβ = φβ, in the C ∗-Algebra U = U(μ), for every value β ∈ R, where β is the parameter that defines the time evolution associated to a homomorphism σt = σβi defined by the potential H . We show that for each real β the KMS state is unique. The probability νβ is the Gibbs state for the potential −β logH . The purpose of the present work is to explain (for an audience which is more oriented to Dynamical System Theory) part the content of a previous paper written by the authors.