Bost-Connes systems, Hecke algebras, and induction

TL;DR

This paper explores the relationship between Bost-Connes systems and Hecke algebras over number fields, demonstrating how the former can be derived from the latter through induction, and establishing a functorial framework connecting number fields and C*-dynamical systems.

Contribution

It introduces a Hecke algebra associated with the affine group over a number field and shows how Bost-Connes systems are obtained via induction, extending the construction to a functorial setting.

Findings

The Hecke algebra is a full corner in the Bost-Connes C*-algebra for K.

A phase transition theorem is derived for the Hecke algebra.

The construction extends to a functor from number fields to C*-dynamical systems.

Abstract

We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field K and we show that the C*-algebra of the Bost-Connes system for K can be obtained from our Hecke algebra by induction, from the group of totally positive principal ideals to the whole group of ideals. Our Hecke algebra is therefore a full corner, corresponding to the narrow Hilbert class field, in the Bost-Connes C*-algebra of K; in particular, the two algebras coincide if and only if K has narrow class number one. Passing the known results for the Bost-Connes system for K to this corner, we obtain a phase transition theorem for our Hecke algebra. In another application of induction we consider an extension L/K of number fields and we show that the Bost-Connes system for L embeds into the system obtained from the Bost-Connes system for K by induction from the group of ideals in K to the group of ideals in L. This gives a C*-algebraic correspondence from the Bost-Connes system for K to that for L. Therefore the construction of Bost-Connes systems can be extended to a functor from number fields to C*-dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMS-states and we show that for beta>1 certain extremal KMS_beta-states for L can be obtained, via induction and rescaling, from KMS_{[L:K]beta}-states for K. On the other hand, for 0<beta\le1 every KMS_{[L:K]\beta}-state for K induces to an infinite weight.