16 OPERATOR ALGEBRASAND DYNAMICAL SYSTEMS
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
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
[ConFW].
The Classification of factors analogous to Theorem 1.5 is the combination
of results (mostly) due to A. Connes, W. Krieger, and U. Haagerup.
1.7.
THEOREM
[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
map
0:L°°(Rr)

Loc(G'//
), 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]).
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