By Adrian Ocneanu

ISBN-10: 3540156631

ISBN-13: 9783540156635

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**Additional resources for Actions of Discrete Amenable Groups on von Neumann Algebras **

**Example text**

Submodel G in ~. For each is a II 1 h y p e r f i n i t e s u b f a c t o r of We call (~, (Ug)) the submodel n, g ~ = ~n ® ((~n), N ~) on w h i c h and (Ad Ug) Ad Ug acts the action. We let R be a c o u n t a b l y infinite tensor p r o d u c t of copies of the 31 submodel factor for each g • G, ~, taken with respect to the normalized we let ag(0) be the c o r r e s p o n d i n g copies of the submodel action factor and Connes (a~ °)) Ad Ug. is an action [3] is free. 8 of and ~(0) : G ÷ Aut R the model action.

5 we have Let us prove (4). g @i gh e G n , g ~I. is easy to obtain from (2), since 7en+ 1 + 7Sn+ 2 + ... < e n . n+l I T ~ 7e n lug - Ug Statement K9 i g e In view of the fact that Proof. is the goal of done before. exist in l'IT-norm and yield a faithful into which U n as the g of G. = and Isnl -I E ]S~I l { k e K g l £ n ( k ) i6I I g k E K~i N g-1 K ni , then =k}l In(k) = g k ~ k. g Since (Sn,Gn)-invariant, • (U~) ~ Isnl -I ~ IS~I e n IK~I i eI n = en n Let us now prove (3). Let g,h,gh e G n.

If all Ei, k in N' o M~ are If not, a tower base of mutually such that IEi, k - E i , k l ~ ~ b E ,- b E 3e½(b E, - b E ) < 36e-i (bE , - b E ) for The idea of the p r o o f lemma. E' = (EL, k ) zero, then a t o w e r we c h o o s e among and then c o n s t r u c t of g e A. (i) and (2) is the following. e. that Ei,kf E' be signifi- be small w i t h respect to this is a c h i e v e d by an adequate choice of the tower basis. (2) we should care that be equivariant, aE, which measures the failure of does not increase too much.

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

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