We present a C\*-algebra which is naturally associated to the ax + b\-semigroup over ℕ. It is simple and purely infinite and can be obtained from the algebra considered by Bost and Connes by adding one unitary generator which corresponds to addition. Its stabilization can be described as a crossed product of the algebra of continuous functions, vanishing at infinity, on the space of finite adeles for ℚ by the natural action of the ax + b\-group over ℚ.

C*-algebras associated with the <var>ax + b</var>-semigroup over ℕ

Joachim Cuntz

Abstract