The ideal structure of the Hecke $C^*$-algebra of Bost and Connes

Abstract.

We compute explicitly the primitive ideal space of the Bost-Connes Hecke $C^*$-algebra by embedding it as a full corner in a transformation group $C^*$-algebra and applying a general theorem of Williams. This requires the computation of the quasi-orbit space for the action of $\Q^*_+$ on the space ${\mathcal A}_f$ of finite adeles. We then carry out a similar computation for the action of $\Q^*$ on the space ${\mathcal A}={\mathcal A}_f \times{\mathbb R}$ of full adeles.