Phase transition in the Connes-Marcolli GL2-system

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 β\ne 0,1. In particular, we show that for each β\in(1,2] there exists a unique KMS_β-state.

23 pages; minor corrections, final version

References in corpus (2)

• From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices
• Modular Hecke Algebras and their Hopf Symmetry

Cited by in corpus (11)

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• Type III_1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers
• Von Neumann algebras arising from Bost-Connes type systems
• Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras
• Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra
• Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers
• Hypergroupoids and C*-algebras
• The Riemann-Roch strategy, Complex lift of the Scaling Site
• On the countable, measure preserving relation induced on an homogeneous quotient, by the action of a discrete group
• The ${\mathrm{GL}}_n$-Connes-Marcolli Systems
• Quantum Statistical Mechanics of $\mathbb{Q}$-lattices and noncommutative geometry