The Regular C*-algebra of an Integral Domain

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®. 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.