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 Â 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 Â .

For the converse we decrease Â , if necessary, and choose a (possibly finite)

sequence (Vn,n 1) in [R] such that B\J[)nVnB = X and Â Ð VnB =

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÷) f° r every g G G and ÷ G G/P, where (L p)(h,x) = h(gh,x). There ex-

ists a Borel set Â c G such that ÃÂ = 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]).