process

C -algebras associated with the ax C b-semigroup over N 205As a consequence of the simplicity of Q N we see that the canonical representationon `2.Z/ bys n . k / D nk ; u n . k / D kCnis faithful. Similarly, if we divide the C -algebra generated by the analogous isometrieson `2.N/, discussed in Section 2, by the canonical ideal K, we get an algebraisomorphic to Q N .The subalgebra F is generated by u and the projections e n , thus by a weightedshift. Thus we recognize F as the Bunce–Deddens algebra of the type where everyprime appears with infinite multiplicity. This well known algebra has been introducedin [3] in exactly that form. It is simple and has a unique tracial state. It can also berepresented as inductive limit of the inductive system .M n C.S 1 // with mapsM n .C.S 1 // ! M nk .C.S 1 //mapping the unitary u generating C.S 1 / to the k k-matrix010 0 ::: u1 0 ::: 0B0 1 ::: 0C@ ::: A :0 ::: 1 0From this description it immediately follows that K 0 .F / D Q and K 1 .F / D Z.Remark 3.5. (a) Just as O n is a crossed product of a UHF -algebra by N, we see thatQ N is a crossed product F Ì N by the multiplicative semigroup N . The algebraQ N also contains the commutative subalgebra D. Since D is the inductive limit of theD n , we see from Lemma 3.3 that Spec D D OZ WD Q O pZ p . The Bost–Connes algebraC Q [1] can be described as a crossed product C. OZ/ Ì N and is a natural subalgebraof Q N (using the natural inclusion C. OZ/ ! F ).We can also obtain Q N by adding to the generators n ;n 2 N (our s n ) and e , 2 Q=Z for C Q , described in [1], Proposition 18, one additional unitary generator usatisfyingue D e u and u n n D n u(here we identify an element of Q=Z with the corresponding complex number ofmodulus 1).(b) The algebra generated by s n (for a single n) and u has also been considered in[5]. It has been shown there that this algebra is simple and contains a Bunce–Deddensalgebra.4 The canonical action of R on Q NSo far, in our discussion, the isometries associated with each prime number appear asgenerators on the same footing and it is a priori not clear how to determine, for two