C*- Algebras and Thermodynamic Formalism

TL;DR

This paper explores the connection between C*-algebras, thermodynamic formalism, and Gibbs states, demonstrating the uniqueness of KMS states and analyzing phase transitions in specific dynamical systems.

Contribution

It provides a detailed analysis of KMS states in C*-algebras related to thermodynamic formalism, highlighting their uniqueness and relation to eigen-probabilities, with an example of phase transition.

Findings

Uniqueness of the ta-KMS state in the constructed C*-algebra.

Relation between KMS states and eigen-probabilities of the dual Ruelle operator.

Existence of phase transition for Gibbs states but not for KMS states.

Abstract

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 $\nu_\beta$ 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.