C*- Algebras and Thermodynamic Formalism

We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1051-1082 (2004). We show only the uniqueness of the beta-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $mu$) and its relation with the eigen-probability $ u_beta$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for the Gibbs state. There is no Phase transition for the KMS state.