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

By Adrian Ocneanu

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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~ .

I) above is an average estimate. other types of estimates COROLLARY. of In Below we give that can be derived from it. the conditions o f the theorem we have f o r any g~G (4) For we any ! k~ lag(Ei,k) - Ei,gkl , ~ 10g ~ , i=l ..... N; ~ > 0 a n d any AkC K i with subset8 k E Ki N g Ki . IAil ~< 61Kil, i=l ..... N, have (5) ~" k[ IEi' kl* ~< 6 + 5 s ½ , Proof. ,N, leg(Ei,k ) -Ei,gkl # Summing for all k,~ < k e Ki N g -I K i and Z 6 K i, leg(~k£-1 (Ei,£) -Ei,k) 1% + lagkZ-1 (Ei,£) -Ei,gkl % as above, we infer IKil ~ leg(Ei,k) -Ei,gkl % < where k e A i.

If k E K i with = £~ h(k) = ghk. hk,ghk e K ni then zn£~(k) g So from the (en,Gn)-invarin ance of K i , it follows that (5) I { k E K ni I Zg£~(k) ~ g£nh(k)}l < en IKnl We have Ugn u nh - ungh IT = ~i E (k, s) ~i En(k2,s),(kl,s)En(kl,s),(k,s) -E~3,s),(k,s) T ~ En - En (k,s) (k2's)'(k's) (k3's)'(k's) 28 where i 6 In, moreover, Hence (k,s) • K9l × S ni' kl = Z~(k) ' k 2 = Zn(kl) g in the last m e m b e r we sum only for t h o s e k and k3 = £ngh (k); for w h i c h k~# k 3. (5) y i e l d s (6) n n IUgU h - U g hn l T IS n I-i ~.

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