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

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

2006-11-17

arxiv.org/abs/math/0611541v1

Mathematics

Operator Algebras

14 pages

Scientific paper

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 $\mathbb Q$.

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