By Adrian Ocneanu

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

**Sample text**

We assume 0 < e < ~,6 and choose that the theorem holds with (i') ~ iKil-i (2' [ i,k Let S cc G 18g(Ei,k) - Ei,klT (Ki × Kj)i,j 34e ½, family any subset of S c c G are invariant is a 2e-paving enough. family for We prove first by 16e½ g6A. of subsets of It is easy to see that the family of G × G 2E-paves and A c e G. (2) replaced lak _i (El,z) _Ei,kl T (Ki)ie ~ be an s-paving (e,A)-invariant. in Theorem we need only the fact that for any invariance degree, the given (and not family of subsets of the group s-paved some subset 56 necessarily all subsets) of the group having that invariance degree.

Since is a W*- 33 subalgebra of M Problem. 2 define for Further on w e constructed sequence from into a Te ~ ~ M, restriction to the of ~ Me (xV)~ to Z(M), is a is a f a i t h f u l restriction certain automorphisms M. Suppose of a u t o m o r p h i s m s of M deal with This ( ~ (x9) )v. Me . restriction For of automorphism of = e - lim x V • M , w h e r e ~ of since . yields such that an a u t o m o r p h i s m of w e are Me and given ~ = iim a m of I~(~,M) Me a exists sending Since II~(x~)II~2 this M' n M e = M e ?

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.