The Regular C*-algebra of an Integral Domain

TL;DR

This paper constructs a purely infinite simple C*-algebra from an integral domain with finite quotients, linking it to adele spaces, crossed products, and generalized Bost-Connes systems, revealing new structural insights.

Contribution

It introduces a natural construction of a C*-algebra associated to an integral domain, connecting it to adele spaces and generalized Bost-Connes systems.

Findings

Identifies the stabilization with a crossed product involving adele spaces.

Provides a universal C*-algebra description for generalized Bost-Connes systems.

Establishes a relationship between integral domains and purely infinite simple C*-algebras.

Abstract

To each integral domain R with finite quotients we associate a purely infinite simple C*-algebra in a very natural way. Its stabilization can be identified with the crossed product of the algebra of continuous functions on the "finite adele space" corresponding to R by the action of the ax+b-group over the quotient field Q(R). We study the relationship to generalized Bost-Connes systems and deduce for them a description as universal C*-algebras with the help of our construction.