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 à c G a discrete subgroup. Then the action Ô of
23
à by left translation on G/P is amenable, but à 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)(÷) =
Previous Page Next Page