$C^*$-algebras associated with the $ax+b$-semigroup over $\mathbb N$

TL;DR

This paper introduces a new $C^*$-algebra linked to the $ax+b$-semigroup over natural numbers, highlighting its simplicity, pure infiniteness, and relation to Bost-Connes algebra via added generators.

Contribution

It constructs a novel $C^*$-algebra associated with the $ax+b$-semigroup over $ $, extending the Bost-Connes algebra by incorporating an additional unitary generator.

Findings

The algebra is simple and purely infinite.

Its stabilization is a crossed product involving finite adeles.

The construction generalizes the Bost-Connes algebra.

Abstract

We present a $C^*$-algebra which is naturally associated to the $ax+b$-semigroup over $\mathbb N$. 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 $\mathbb Q$ by the natural action of the $ax+b$-group over $\mathbb Q$.