amenable (take G = SL(2,R), Ñ = {(J°):i e R} , and T=SL(2,Z) : this
action is related to example 1.2(5), and its hyperfiniteness was first proved
in [Bow2] and [Ver2]). As far as I am aware, it is not known whether every
countable infinite group has a free, nonsingular, ergodic, hyperfinite action
on a measure space (×, ì) .
(3) Induced equivalence relations and subrelations. Let R be a nonsingular,
ergodic equivalence relation on (X, SP, ì) , and let  eS* with ì(Â) 0.
Then RB is amenable (or hyperfinite) if and only if R is amenable: indeed,
if (Rn , ç 1) is an increasing sequence of finite subrelations of R with
|J„R„(-x) = R(x) for ì-a.e . ÷ e X, then ((Rn)B, ç 1) is an increasing
sequence of finite subrelations with \Jn{Rn)B(x) = BB(x) for ì-a.e . ÷ e  .
For the converse we decrease  , if necessary, and choose a (possibly finite)
sequence (Vn,n 1) in [R] such that B\J[)nVnB = X and  РVnB =
Fm5 ç KnÄ = 0 for all m ö n. The maps {F„ , ç e {1, ...}} define an
isomorphism É ~ à ÷ { 0 , 1 , . . .} which carries R to R5 ÷ R', where R'
is the transitive equivalence relation on {0, 1, ...} , and the hyperfiniteness
of RB implies the hyperfiniteness of R.
If R is amenable and S c R is a subrelation, then S is again amenable:
choose finite subrelations (Rrt , ç 1) with Rn{x) / R(x) ì-a.e. , and note
that Sn(x) / S(x) ì-a.e. , where Sw = R
n S , ç 1.
(4) Amenability of equivalence relations on Markov shifts. All the equiva-
lence relations in example 1.2(3)—(4) are amenable by Corollary 12 in
The Classification of factors analogous to Theorem 1.5 is the combination
of results (mostly) due to A. Connes, W. Krieger, and U. Haagerup.
[Con2], [Haa]. Let Ç be á complex, separable Hubert
space, and let sf c B(H) be afactor. The following conditions are equivalent.
(1) j / isinjective, i.e. thereexistsáprojection E:B(H)—B(H) ofnorm
1 with E{B{H))=stf;
(2) $f is hyperfinite, i.e. there exists an increasing sequence (sfn, ç 1)
of finite dimensional subalgebras of srf whose union is {strongly) dense
in srf;
(3) sf ~ sf(ß) for á hyperfinite equivalence relation R on á measure
space (X, SP, ì) .
Finally, if R and S are hyperfinite, ergodic equivalence relations on measure
Spaces (X, S?, ì) and (Y, ^~, v), respectively, then s/(R) and J / ( S ) are
isomorphic if and only if R and S are isomorphic, i.e. if and only if there
exists an isomorphism Ö:sf(R) -+sf(S) such that Ö(^^)) = Jt(S).
*À{ø)(8÷) r every g G G and ÷ G G/P, where (L p)(h,x) = h(gh,x). There ex-
ists a Borel set  c G such that à= G and yB Ð Â = 0 for 1 ö ã e Ã, and the

), given by è (ø) = ç(ø') with y/\g, ÷) = ø(ã÷, x) whenever
÷ G G/P , ã G Ã, g G yB , is a left invariant mean on R (cf. [Zim6]).
Previous Page Next Page