Phase transition in the Connes-Marcolli GL2-system

TL;DR

This paper develops a framework for analyzing KMS-states on C*-algebras from Hecke pair actions, classifies these states for a specific system, and finds unique states for certain inverse temperatures.

Contribution

It introduces a general method for studying KMS-states on Hecke pair C*-algebras and applies it to classify states in the Connes-Marcolli GL2-system.

Findings

Unique KMS_eta-states for eta in (1,2]

Classification of KMS-states for eta ≠ 0,1

Framework applicable to Hecke pair C*-algebras

Abstract

We develop a general framework for analyzing KMS-states on C*-algebras arising from actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli and classify its KMS-states for inverse temperatures \beta\ne 0,1. In particular, we show that for each \beta\in(1,2] there exists a unique KMS_\beta-state.