Entropy theory for the parametrization of the equilibrium states of Pimsner algebras

TL;DR

This paper develops an entropy-based framework to classify equilibrium states of Pimsner algebras derived from finite rank C*-correspondences, identifying critical temperatures and unifying existing classification methods.

Contribution

It introduces an entropy theory that determines the critical inverse temperature for equilibrium states and unifies various classification approaches for Pimsner algebras.

Findings

Infimum of tracial entropies dictates critical inverse temperature.

Entropy of the ambient C*-correspondence influences equilibrium states.

When the diagonal is abelian, strong entropy defines the maximum critical temperature.

Abstract

We consider Pimsner algebras that arise from C*-correspondences of finite rank, as dynamical systems with their rotational action. We revisit the Laca-Neshveyev classification of their equilibrium states at positive inverse temperature along with the parametrizations of the finite and the infinite parts simplices by tracial states on the diagonal. The finite rank entails an entropy theory that shapes the KMS-structure. We prove that the infimum of the tracial entropies dictates the critical inverse temperature, below which there are no equilibrium states for all Pimsner algebras. We view the latter as the entropy of the ambient C*-correspondence. This may differ from what we call strong entropy, above which there are no equilibrium states of infinite type. In particular, when the diagonal is abelian then the strong entropy is a maximum critical temperature for those. In this sense we complete the parametrization method of Laca-Raeburn and unify a number of examples in the literature.