Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

Abstract

We consider the Hecke pair consisting of the group $P_K^{+}$ of affine transformations of a number field $K$ that preserve the orientation in every real embedding and the subgroup $P_O^{+}$ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra $C^{\ast }(P_K^{+},P_O^{+})$ has a natural time evolution $\sigma$, and we describe the corresponding phase transition for KMS${}_{\beta }$-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated to $K$ has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of $C^{\ast }(P_K^{+},P_O^{+})$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^{\ast }(P_K^{+},P_O^{+})$ on which ground states exhibit the `fabulous' property with respect to an action of the Galois group $Gal(K^{ab}/H_{+}(K))$, where $H_{+}(K)$ is the narrow Hilbert class field. In order to characterize the ground states of the $C^{\ast }$-dynamical system $(C^{\ast }(P_K^{+},P_O^{+}),\sigma )$, we obtain first a characterization of the ground states of a groupoid $C^{\ast }$-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.

arXiv comment21 pages; v2: minor changes and corrections

References in corpus (16)

• KMS states of quasi-free dynamics on Pimsner algebras
• KMS states on the C*-algebras of non-principal groupoids
• On Bost-Connes type systems for number fields
• The Regular C*-algebra of an Integral Domain
• Phase transition in the Connes-Marcolli GL2-system
• Bost-Connes systems, Hecke algebras, and induction
• Type III_1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers
• Phase transitions on Hecke C*-algebras and class-field theory over Q
• K-theory for ring C*-algebras - the case of number fields with higher roots of unity
• Induced Representations and Hypergroupoids
• Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties
• Reconstructing global fields from Dirichlet L-series
• Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra
• KMS weights on graph C*-algebras II Factor types and ground states
• Reconstructing the Bost--Connes semigroup actions from K-theory
• Reconstructing global fields from dynamics in the abelianized Galois group

Cited by in corpus (4)

• The groupoid approach to equilibrium states on right LCM semigroup C*-algebras
• Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers
• Partition functions as C*-dynamical invariants and actions of congruence monoids
• Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras