1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

178 5. Exact Groups and Related Topics

This implies

r !ri(x) dm(x) < r I: 11s.c - (^3 ·x11 dm(x) +

8

; < s

Jx JxsEE

and we obtain the claim.

Now, let a finite subset E c rands> 0 be given. By our work above,

there exists a continuous map 'fJ such that supxEX lls.rJx _'f/s.xll < s for every

s E E. We may assume that there exists a finite subset F c r such that

supp 'f/x c F for all x E X. Picking one point out of every orbit, we can

find a f-fundamental domain V c K - i.e., K decomposes into the disjoint

union LJvEV I'v -and let v: K --* V be the corresponding projection ( v takes

every element in an orbit to its representative in V). Next we fix a cross

section O" : K --t r such that a = O"( a). v (a) for every a E K. Since the map

vis constant along orbits, O"(s.a)-^1 sO"(a) E rv(a) for every s Er and a EK.

For each v EV, we set

Ev= {O"(s.a)-^1 sO"(a): a E F n I'v ands EE} C rv.

Let Y be a compact r-space which is amenable as a rv-space for every
v E V. (Such Y always exists when each ra is exact - take Y = ,Br).
The proof of our proposition will be complete once we see that X x Y is
an amenable f-space (with the diagonal action). Indeed, this will imply I'
is exact; moreover, if we can take Y = X, then the diagonal embedding
X <--+ X x X is r-equivariant and continuous - hence amenability of X will
follow from amenability of the diagonal action on X x X. Since Y is rv -
amenable and Ev is finite, there exists a continuous map vv: Y-- Prob(r)
such that
max sup lls.v~ - vi·Yll < s.
sEEv yEY
Now, we defineμ: Xx Y-- Prob(r) by
μx,y = L TJx(a) O"(a).v:c~(1·Y.
aEF

The mapμ is clearly continuous. Moreover, we have

s.μx,y = L TJX( a) SO"(a).v:c~rl·Y

aEK

= L TJx(a) O"(s.a).(O"(s.a)-1SO"(a).v:c~(1·Y)

aEF

pjc L TJx(a) O"(s.a).v:c~:(1s.y

aEK

pjc 6 '"°"' 'f/ s · x( s.a ) ( ) O" s.a .vv(s.a) u(s.a)-^1 s.y

aEK