process

212 J. CuntzThe operator s 1 plays a somewhat special role and we therefore denote it by f . Thenthe s n ;n2 Z and u generate the same C -algebra as the s n ;n2 N together with u andf . The element f is a selfadjoint unitary so that f 2 D 1 and we have the relationsfs n D s n f; f uf D u 1 :We consider now again the universal C -algebra generated by isometries s n ;n 2 Zand a unitary u subject to the relations (1). We denote this C -algebra by Q Z . We seefrom the discussion above that we get a crossed product Q Z Š Q N Ì Z=2 where Z=2acts by the automorphism ˛ of Q N that fixes the s n ;n2 N and ˛.u/ D u 1 .Theorem 7.1. The algebra Q Z is simple and purely infinite.Proof. Composing the conditional expectation G W Q N ! D used in the proof ofTheorem 3.4 with the natural expectation Q Z D Q N Ì Z=2 ! Q N we obtain again afaithful expectation G 0 W Q Z ! D. The rest of the proof follows exactly the proof ofTheorem 3.4, using in addition the fact that the f i in that proof can be chosen such thatf i ff i D 0.Denote by P Q the full ax C b-group over Q, i.e. 1 bP Q D j a 2 Q ;b2 Q :0 aTheorem 7.2. Q Z is isomorphic to the crossed product of C 0 .A f / or of C 0 .R/ by thenatural action of P Q .Proof. This follows from Theorems 6.4 and 6.5 since P Q D P C Q Ì Z=2.On Q Z we can define the one-parameter group . t / by t .s n / D n it s n ;n 2 N .The fixed point algebra is the crossed product F Ì Z=2 of the Bunce–Deddens algebraby Z=2.In order to compute the K-groups of Q Z we first determine the K-theory for F 0 DF Ì Z=2. This algebra is the inductive limit of the subalgebras A 0 n D C .u; f; e n /.Lemma 7.3. (a) The C -algebra C .u; f / is isomorphic to C .D/, where D is thedihedral group D D Z Ì Z=2 (Z=2 acts on Z by a 7! a). For each n D 1;2;:::,the algebra A 0 n is isomorphic to M n.C .D//.(b) We have K 0 .A 0 n / D K 0.C .D// D Z 3 and K 1 .A 0 n / D K 1.C .D// D 0 forall n. The generators of K 0 .C .D// are given by Œ1 and by the classes of the spectralprojections .uf / C and f C of uf and f , for the eigenvalue 1.(c) Let p be prime. If p D 2, then the map K 0 .A 0 n / ! K 0.Apn 0 / is described, withrespect to the basis Œ1, Œ.uf / C and Œf C of K 0 .C .D// Š Z 3 , by the matrix02 110@ 0 0 1A0 0 1