process

C -algebras associated with the ax C b-semigroup over N 213If p is odd, then the map K 0 .A 0 n / ! K 0.Apn 0 / is described by the matrix0@ p p 1 1p 12 20 0 1 A :0 1 0Proof. (a) It is clear that the universal algebra generated by two unitaries u; f satisfyingf 2 D 1 and fuf D u is isomorphic to C .D/.In the decomposition of C .u; f; e 2 / with respect to the orthogonal projections e 2and ue 2 u 1 , the elements u and f correspond to matrices 0w1 0and f0 00 f 1where w is unitary and f 0 ;f 1 are symmetries (selfadjoint unitaries).The relations between u and f imply that wf 1 D f 0 and that wf 1 is a selfadjointunitary, whence f 1 wf 1 D w . Thus A 0 2 is isomorphic to M 2.C .w; f 1 // and w; f 1satisfy the same relations as u; f .If p is an odd prime, then in the decomposition of C .u; f; e p / with respect to thepairwise orthogonal projections e p ;ue p u 1 ;:::;u p 1 e p u .p 1/ , u and f correspondto the following p p-matrices0101 f0 0 ::: w0 0 ::: 01 0 ::: 00 0 ::: f 1B0 1 ::: 0C@ ::: A and 0 0 ::: f2 0B :::C@ 0 0 f0 ::: 1 02 ::: A0 f 1 ::: 0 0where w and the f 1 ;:::;fp 12are unitary and f 0 is a symmetry.The relation fuf D u implies that f 0 D wf 1 , f 1 D f 2 DDfp 12and f 2i1 for all i. From .wf 1 / 2 D 1 we derive that f 1 wf 1 D w . Thus Ap 0 Š M p.C .w; f 1 //and w; f 1 satisfy the same relations as u; f .The case of general n is obtained by iteration using the fact that C .u; f; e nm / ŠM m .C .u; f; e n //.(b) This follows for instance from the well known fact that C .D/ Š .C ? C/ .(c) Let p D 2. Using the description of K 0 .C .u; f // under (b) and the descriptionof the map C .u; f / ! C .u; f; e 2 / from the proof of (a), we see that the generatorsŒ1, Œ.uf / C and Œf C are respectively mapped to 2Œ1, Œ1 and Œf C C Œ.uf / C .Let p be an odd prime. The matrix corresponding to f in the proof of (a) is conjugateto the matrix where all the f i , i D 1;:::;.p 1/=2 are replaced by 1. Thus the classof f C is mapped to Œ.wf 1 / C C p 12 Œ1.The matrix corresponding to uf is conjugate to a selfadjoint matrix of the sameform as the matrix representing the image of f , but with the upper left entry f 0 replacedby f 1 . Thus the class of .uf / C is mapped to Œf C1 C p 12 Œ1.D