OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS

17

In order to see a connection between injectivity and amenability we as-

sume that R is a nonsingular, ergodic equivalence relation on a measure

space (×95",ì), and that sf(R) is injective. We set Ç =

L2(R,

ì^) and

assume that E{B(H)) = s/(R) for a projection E:B{H) -+ B(Z7) of norm

1. Then Å is a conditional expectation [Tom], i.e. E(ABC) = AE(B)C for

all Á, C G J/(R) and £ G B(JJ). In particular, if / G L°°(R, ì^) (acting

on // by multiplication), then E(f) G J/(R), and MhE(f) = E(Mhf) =

E{fMh) = E(f)Mh, so that E(f) G Jt(R)'. Since ^T(R) is maximal

Abelian in J / ( R ) we see that E(f) e J?{R). Finally, E{Lv-xfLv) =

Lv-xE(f)Lv = E(f) ·

V~l

for every V G [R], so that / - £(F) is a

left invariant mean on R (cf. [ConFW]). Conversely, if R is amenable, then

Theorem 1.5 implies that R is hyperfinite, and it follows quite easily that

J / ( R ) is again hyperfinite. The equivalence of hyperfiniteness and amenabil-

ity is intimately related to Rokhlin 's lemma, which we shall discuss in Section

3.

One of the main ingredients in the proof of the exact correspondence be-

tween (isomorphism classes of) hyperfinite ergodic equivalence relations and

hyperfinite factors in the last Statement of Theorem 1.7 is the Classification of

hyperfinite equivalence relations by W. Krieger [Kril], [Kri2], [Kri3], [Kri4].

The ideas involved in this Classification bring us to our next topic, cohomol-

ogy.