By Adrian Ocneanu

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P O1,3.. ~< k = kn(k,i) (Z,t) 6 E IKnl (ILn,jl ITn . 3(3) _n+l t;(~2,{),(k,~) E K~ × L~ • × Tn . E l{(k,Z) • Kn×Knll,ji~n(£g(k),i) # %g+l(~n(k,i)) } IT~,jlnisnll~n+ll-1 1,3 ~< 2"3sn ~ l~il ILni,jl l~i,jl ISnl Isn+ll-1 l, 3 = 6s n Finally, . n . n+l, IU -U , , IT ~ IZI-ZIIT + ]Z21T + I~21T ~ 6~n+E n = 7Cn and (2) is proved. Let A n be the maximal abelian subalgebra of ~n generated by n ~ • ~n. Then An C An+l and if we let A denote the weak (E~,~), closure of u A n in ~, then A is a maximal abelian subalgebra in ~.

For all Proof. 1 (if we take e. The idea of the proof of of any Me . The estimate being Me Bg = e(l,g ) then (Ei, k) can be chosen in the relative commutant in M e Moreover in sets in is to take Rohlin G ×G, towers indexed by and then sum after the coordinate. Step A. 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.

5 The R o h l i n maximality Theorem argument. the set of f a m i l i e s tions in N'N M e E= (Ei,k)iei a E < 3e ½ b E (2) Cg,E < 3@e-lbE is n o n v o i d E E