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

By Adrian Ocneanu

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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. for and lek~g(ek) I~ < Ylekl ~ We have gE K . ,q ; sub C*-algebra ek A N, Let '" of ~ ( I~, M~) Z~(IN,M w) is kept globally term by term on yielded then ek ~(~,M~)). ,q, geE. 48 We also have for all Ad Um(e k) and thus e k E N' N M~. 4 or M~ Ad Um(~(ek)) = ~(ek Mm, we shall part by E= G apply bE = 1 ~ k=0 the Index in o r d e r of the p r o o f .....

To p r o v e * - s u b s e t of s c a l a r values; o u t e r and let Then q' < 1 - (q V ~(q) V B-I(q)) it we i n f e r Let takes < ¼ T(q). (q' V 8(q'))(qV 8(q)) = 0 dicted is, by T x E M e. ~(q~(q)) if not, same reasoning for 8 e Aut M such t h a t Indeed, then is a factor, [x[T = ~(Ix[) maximal q' ~ 0 M (~n)n be a t o t a l sequences ~E~ that B = (~)e there exists We r e m a r k fixed. for a By m e a n s e Aut M e a projection t h a t in the a l g e b r a Me we have T~(18(q)aV-a~qI2 ) = Te(] (8(q)-q)aV] 2) = Te(laVlZ)T((8(q)-q)) 2 > ~/~ Te(laVl2) H e n c e we can p i c k o u t of a r e p r e s e n t i n g element q~e M such that llqV I[ <~ 1 (q~)~ r e p r e s e n t s Ho~(q)a-aq[l and the c o n t r a d i c t i o n q an >i 1/21ja~ll~ , 1 ll[q ,Sk]ll# ~< U T h e n the s e q u e n c e for and II~(q~)a~ -a~q~)I1# 1 li[qv,~k]N <~ ~ sequence 2 i> thus o b t a i n e d k=l ' .....

Such that for any 05 = W - lim T (x n) n÷t0 if X n = X for all = ~(~(x)) sequences automorphisms leave it globally ~: e ÷ k = (Xn) n E d (i) part lemma. sequence. M ~, which there w E V n \ V n + 1 , and for (xk(V)) w. to the one of the p r e c e d i n g w e can p u t t o g e t h e r predual if by (a(Xn)) n of . 37 Remark. then From Proof. = u Cn n invariant Let a unital by ~, and M n • M n + l • M, sequence any (5) ~ • M, , Let Vn, be n~ 1 such be finite sets w i t h on M. n) Yp(n) (9) lla~(Xp(n))- for all The lemma and now In w h a t follows, i/n as in the such , x= union M,.

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