By Adrian Ocneanu

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5. n Remark. The above inequality states that the proportion li,j of right translates of K9 in K~ +I almost doesn't depend on j. This l 3 n j is in fact arbitrary. What might be quite surprising since li, actually happens is that the ergodic measure ~ and the level refinement n j is almost indepen"choose" a part of the system (K ni)n,i for which li, dent of j ; on the rest of the diagram, ~ being small, the contribution of the corresponding terms in the sum (2) is negligible. × Tgl,j and I l,jl : min{l -n R~ P~ .

LJyll~ i, we have Y with then = (x~)v E Me, then (xV)~. Thus P III -~ lim induces to the one induced by the inclusion. The lemma is proved. ~ ei,j[x, ej, i] i,j (P(x V) - x v) = 0 a map M~ ÷ *-strongly and so (E'A M)~ that is inverse 41 C h a p t e r 6: THE ROHLIN T H E O R E M In this chapter we prove a Rohlin type t h e o r e m for a d i s c r e t e a m e n a b l e group G As a consequence, a c t i n g c e n t r a l l y freely on a yon Neumann algebra. 1 G/H G, the on the almost H. Some of the basic tools in the m o d e r n d e v e l o p m e n t s of the ergodic theory in b o t h m e a s u r e spaces and von N e u m a n n algebras are the various e x t e n s i o n s of the Rohlin Tower Theorem.

Is faithful. to the faithful extension of [i, If M~ (M), state on are W*-subalgebra8 and is faithful. M. Let @~(x*x) = 0 then x>0 in (xW)w 6 ~ By means of the GNS construction we may suppose a separating that M~ is a C*-sub- cyclic vector 6. We show (Mm) h is so-closed. 1 Let (xi) i C (M~)~ fundamental xy 6 H (Me)~ with to some complete, be a so-fundamental in the topology M For any y, is so-closed xE xiY ~ ÷ xYy~. and so x e(MW)~; net; of the lemma before, xi~ ÷ xY~, does not depend on of $ % be a faithful (M~) + be represented that is a W*-algebra For any faithful Proof.