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

TL;DR

This paper analyzes the phase transitions and ground states of a Hecke algebra associated with affine transformations of a number field, connecting number theory, operator algebras, and Galois actions.

Contribution

It characterizes ground states of a groupoid C*-algebra independently of number theory and relates this to the structure of Hecke algebras and arithmetic subalgebras.

Findings

Describes phase transition for KMS states and ground states.

Identifies an arithmetic subalgebra with the 'fabulous' property.

Provides a groupoid C*-algebra ground state characterization.

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 $C^*(P^+_K,P^+_O)$ 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 $C^*(P^+_K,P^+_O)$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^*(P^+_K,P^+_O)$ 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 $C^*$-dynamical system $(C^*(P^+_K,P^+_O),\sigma)$, we obtain first a characterization of the ground states of a groupoid $C^*$-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.