process

C -algebras associated with the ax C b-semigroup over N 203Lemma 3.2. In Q N we have(a) e n D P m 1iD0 uin e nm u in for all n; m 2 N ;(b) e p s q D e pq s p D s q e p and e p e q D e pq D e q e p when p and q are relativelyprime;(c) sn s m D s m sn for n, m relatively prime.Proof. (a) This follows by conjugating the identity 1 D P u i e m u i by s n t sn andusing the fact that s n e m sn D e nm.(b) Since the u i e pq u i are pairwise orthogonal for 0 i < pq, we see thatu lp e pq u lp ? u kq e pq u kq if 0