KMS States, Entropy and a Variational Principle for Pressure

TL;DR

This paper explores the relationship between entropy, pressure, and KMS states in $C^*$-Algebras, introducing a natural entropy definition based on transfer operators and establishing a variational principle linking KMS states to equilibrium measures.

Contribution

It introduces a new, simple entropy definition for $C^*$-Algebras using transfer operators and connects KMS states with thermodynamic formalism through a variational principle.

Findings

Defined a natural entropy for $C^*$-Algebras via transfer operators

Established a variational principle for pressure as a min-max problem

Linked KMS states to equilibrium measures in the context of continuous transformations

Abstract

We want to 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 describe here is quite natural and extends the usual one for Dynamical Systems in Thermodynamic Formalism Theory. It has the advantage of been very easy to be introduced. It is basically obtained from transfer operators (also called Ruelle operators). Later we introduce the concept of pressure as a min-max principle. Finally, 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 KMS states for certain algebras coming from a continuous transformation and equilibrium measures.