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

P(X) IiI-li,j = an inclusion (E'~ M)~ ÷ M~.

__> 2 -n+l xi therefore in As converges M~ ~ for any y 6 M m, and so there is separating, on the dense is xy subset is a W*-algebra. M e , and hence xi is Being M~ ll'iJ% is a W*-subalgebra. (M~) h we have N [x,y]ll~ < 211xJt$ llyll + I1xyli~ + 11xy*II~ The left members they vanish precisely are thus so-continuous for seminorms x 6 ( M ) h we have proved that for on M yEM (M~) h. 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.

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,. Let a representing (xW)~, lemmas real above. that (Xm) m • C n, V n D _ V n + I, be d e f i n e d ~(x) • M e in Me by , ~ • F n.

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