OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS

15

relation on a measure Space (X, S?, ì) . Á left invariant mean on R is a

map P:L°°(R, ì ê) — • L°°(X, ì) with the following properties: P(l) = 1,

/(/) 0 whenever / 0, and P{Lvf) = {Pf) ·

V~l

for every / G

L°°(R, ì

ê

), Ê G [R], where Lvf(x, x') =

f(V~lx,

x'). The equivalence

21

relation R is amenable if it admits a left invariant mean.

The main result in [ConFW] states that, for a nonsingular equivalence rela-

tion R, amenability is equivalent to the notion of hyperfiniteness introduced

in [Dyel].

1.5.

THEOREM

[Dyel], [ConFW]. Let R be á nonsingular equivalence rela-

tion on á measure space (X, S?, ì). The following conditions are equivalent.

(1) R is amenable;

(2) R is hyperfinite, i.e. there exists an increasing sequence (Rw , ç 1)

offinite subrelations of R such that R(JC) = UrtRw(x) for ì-a.e.

xeX;

(3) R =

Rv

= {(x9 V x):x G X, k e Z} for á nonsingular automor-

phism V ï/(× 9&,ì).

1.6.

EXAMPLES.

(1) Actions of amenable groups [Ziml]. Let Ô be a non-

singular action of a countable, amenable group G on a measure space

(X, S?, ì) . If R =

RT

is defined as in example 1.2(1) then R is amenable.

Indeed, let m be an invariant mean on G, and let / G L°°(R, ì

ê

). For

every ÷ e X we define a surjective map ^x:(? — • {{x, x):x G R(.x)} by

px(g) = (JC, Ã^÷). Then fx · ö÷ e l°°(G), where /^ denotes the restriction

of / to {(JC, x')\x G R(x)}, and we set F(x) = m(fx · ö÷). The map

f — F = P(f) is a left invariant mean on R.

(2) Amenable actions of nonamenable groups [Zim2], [Zim3]. If an ac-

tion Ô of a countable group G is measure-preserving and free, then G is

amenable if and only if R

r

is amenable [Ziml]. However, a nonamenable

ô

group may have a free nonsingular amenable action Ô (i.e. R is amenable):

let G be a locally compact second countable group, Ñ c G an amenable

closed subgroup, and Ã c G a discrete subgroup. Then the action Ô of

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Ã by left translation on G/P is amenable, but Ã need obviously not be

21A

right invariant mean on R is a map P':L°°(R, /zR) - L°°(^, ì) with ^'(1) = 1 ,

P\f) 0 if / 0 , and P'(Rvf) = P'(f) · F "

1

for all / e L°°(R, ì Ë) and Ê e [R], where

{Rvf)(x,x) = f(x,

V~xx).

If Ñ is a left invariant mean on R then / -• /'(/) = P(f · È)

is a right invariant mean, where è(÷, x) = (x , x) is the flip on R.

Á locally compact, second countable group G is amenable if there exists an invariant mean

on L°°(G), i.e. a linear functional m:L°°(G) —• C with m(l) = 1 , m(/) 0 whenever /

0 , and m(fg) = m(f) =

m(fg)

for all / € L°°(G), g e G , where fg{h) = f(gh) = /*(*) .

Let ì be a probability measure on G/P which is quasi-invariant under G, and let

m be an invariant mean on Ñ. Define ?/:L°°((J ÷ G/P) -+ L°°(G/P) by setting //(#) =

m{f ö(·, x)dß(x)) and observe that ^ is linear, ?y(^) 0 for #? 0, and ^L ö)(÷) =