By Adrian Ocneanu

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

**Example text**

Thus (5) is obtained from Here are some circumstances theorem is fulfilled: If the algebra M needed, since #~ under which the hypothesis is a factor, is the canonical semiliftable automorphisms. In the case when ~: G ÷ Aut M crossed action action; G ÷ Aut M, for instance if M (1). 7, is the hyperfinite ~ is a strongly free II 1 of I I factor, then • 43 any free action G ÷ Aut M is c e n t r a l l y free. 2 ~ M = L (X, ~, ~), with ~ a probability is induced by a measure p r e s e r v i n g free action of G on is strongly free and one gets the O r n s t e i n and Weiss theorem.

We show that for any f E P r o j ( N ' n M there exists e E Proj (N'n Mw) with (5) e ~< f (6) Ieag (e) I~ < YIel~ ' g e K (7) i E {i ..... m} , ) and any y > 0 lel~ <~ (l+ IKI)-I Ifl~. The family of projections e e N'A M~ satisfying (5) and (6) is nonvoid and well ordered, so let e be maximal with these properties. We show that (8) e e V( g also satisfies eV-K~g(e))V (l-f> = 1 If not, let e' be a nonzero projection in N'A M e orthogonal to the left member of (8). By Step A there exists a nonzero projection e" in N'e S~, e"<~e', with Ie"~g(e") I~ < 811e"I~ , g e K.

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 ' ..... k,~ = i, an e l e m e n t I/2J]al] z T shows ~ .