process

204 J. CuntzConsider the action of T n on B n given by˛.t1 ;:::;t n /.s pi / D t i s piand denote by F n the fixed-point algebra for ˛ (i.e. F n D F \ B n ). There is a naturalfaithful conditional expectation E W B n ! F n defined by E.x/ D R T n ˛t.x/dt.Now D n is the fixed point algebra, in F n , for the action ˇ of T on F n given byˇt.e k / D e k ;ˇt.u/ D e it uand there is an associated expectation F W F n ! D n defined by F.x/ D R T ˇt.x/dt.The composition G D F ı E gives a faithful conditional expectation A n ! D n .These conditional expectations extend to the inductive limit and thus give conditionalexpectations E W Q N ! F , F W F ! D and G W Q N ! D.Theorem 3.4. The C -algebra Q N is simple and purely infinite.Proof. Since inductive limits of purely infinite simple C -algebras are purely infinitesimple [8], 4.1.8 (ii), it suffices to show that each B n is purely infinite simple.For each N , denote, as above, by D n.N / the subalgebra of D n generated byfu k e l u k ;k2 Zg where l D p1 N pN 2 :::pN n . The natural map Spec D n ! Spec D n.N /is surjective and, by the proof of Lemma 3.3, D .N /n Š C l .Choose 1 ; 2 ;:::; l in Spec D n such that f 1 ; 2 ;:::; l g!Spec D n.N / is bijectiveand such that i .s k t s k / ¤ i.t/for all k of the form k D p m 11:::p m nn with m i N . We can choose pairwise orthogonalprojections f 1 ;f 2 ;:::;f l in D n with sufficiently small support around the i such thatf i s k f i D 0 for all 1 i l and k as above and such that f i xf i D i .x/f i for allx 2 Dn N /. Then the map ' W D.N n ! C .f 1 ;:::;f l / Š C l defined by x 7! P f i xf iis an isomorphism.We denote by P .N / the set of linear combinations of all products of the formu k s m 1p 1:::s m np nwith m i N .Forx 2 P .N / we have that'.G.x// D X f i xf i :Let now 0 x 2 B n be different from 0. Since G is faithful, G.x/ ¤ 0 and wenormalize x such that kG.x/k D1. Let y 2 P .N / , for sufficiently large N , be suchthat kx yk