The Radon-Nikodym problem for approximately proper equivalence relations

TL;DR

This paper investigates the Radon-Nikodym problem for approximately proper equivalence relations, focusing on the uniqueness of Gibbs states, and introduces a variant of the dimension group to derive conditions for trace uniqueness and aspects of the Perron-Frobenius-Ruelle theorem.

Contribution

It introduces a new variant of the dimension group and applies it to establish criteria for Gibbs state uniqueness and trace classification in AF algebras.

Findings

Established sufficient conditions for trace uniqueness on AF algebras

Connected the Radon-Nikodym problem to Gibbs state uniqueness

Extended parts of the Perron-Frobenius-Ruelle theorem

Abstract

We study the Radon-Nikodym problem for approximately proper equivalence relations and more specifically the uniqueness of certain Gibbs states. One of our tools is a variant of the dimension group introduced in the study of AF algebras. As applications, we retrieve sufficient conditions for the uniqueness of traces on AF algebras and parts of the Perron-Frobenius-Ruelle theorem.