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# Actions of Discrete Amenable Groups on von Neumann Algebras by Adrian Ocneanu

By Adrian Ocneanu

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Q , which invariant Let Trick. ,q I~ ~ 6 k=l generating be a separable all the projections by the automorphisms ~: e ÷ M W If C (9) and thus prove the 1 be unitaries and let does not with O {Ad U m l m E I N } C Aut M e. 5 to the projec- a family (n) . (n) < 1 e k ~g ~e k ) I~ n ' = {~gigEG} and in detail. let us choose (n) ek = le(k0) Step C is proved. small, to make y = 0 in this procedure N ' N M~ k such that k, thus e0 = fq+ I, can be taken arbitrarily Let us describe N'A M e . ,q, gE K (i + IKI)-I Ifkl~ Ifk+ll#~< ( i - ( i + Step D.

Connes. y E S' n M~ Te(la[ 2) in nontrivial B~(z) - z ~ 0. M, and But Bk(p) IBe(z) -zl 2 # 0. and q = k ~ zBk(P)" B(q) = q, there is qlB~(z) - zl 2 ~ 0 , Let x= (B~k(z); then plBe(x) - x I ~ 0. We n o w use the Slow R e i n d e x a t i o n N the s m a l l e s t contains x, W*-subalgebra and let and the c o u n t a b l e y e S' A Me, F Me S. a(y) We send = B~(Y) Te(la*1218w(y)-yl F r o m our choice that A Let of x ~ = {~,B w} 6 A u t ( M e ) , leaves be the sub W * - a l g e b r a subset ya = a y , of Trick.

F; In v i e w of (Ei, k) to The only p r o b l e m is the fact that we alter the old tower. If f was e - i n v a r i a n t then cutting with l-f w o u l d not affect a E. We approximate this by taking a tower indexed by a very large subset K' of G; such a tower has a very good global invariance, we regroup its p r o j e c t i o n s to get the tower For G=~, I = {i} and (fi,k) and s u b s e q u e n t l y indexed by K I .... ,KN- K l = {1,2 ..... p} c ~, a typical picture w o u l d be the following: X Z mp ......

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