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

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 the multiplicative positive rationals on the space of finite adeles. We then carry out a similar computation for the action of the nonzero rationals on the space of full adeles.