1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

176 5. Exact Groups and Related Topics

map 'T/: Y --->Prob( A) c Prob(I') such that

maxmaxsup ll(cr(p)-^1 s-^1 cr(sp)).'TJY - 'T/o-(p)-is-lo-(sp).yll < s.

sEE pEF yEY

We define μ: X x Y ---> Prob(I') by

μx,y = L C(p)cr(p).rJo-(p)-lY.

pEF

Then, μ is continuous and one checks that

μs.x,s.y = L e·x(q)cr(q).rJO"(q)-ls.y

qEF

~e L~x(S-lq)cr(q).rJo-(q)-ls.y

qEf

= L C(P )scr(p) ( cr(p )-1 s-lcr(sp)) ·'T/o-(sp)-1s.y

pEF

~e L C(p)scr(p).rJo-(p)-1.y

pEF

= s.μx,y

for all s E E and ( x, y) E X x Y.

5.2. Groups acting on trees

D

For any I'-space K, we denote the stabilizer subgroup of a E K by ra =
{ s E r : s .a = a}. Our goal here is to show that a group acting on a tree is
exact whenever all the vertex stabilizers are exact. This gives an alternate
proof of the fact that amalgamated free products of exact groups are exact,
since r = I'1 *A r 2 acts on a tree in such a way that the vertex stabilizers
are conjugates of r 1 or r 2. (See Appendix E.)
Our first result is inspired by, and generalizes, Proposition 5.1.11. The
flexibility provided by Borel maps makes it very useful.

Proposition 5.2.1. Let r be a countable group, X a compact I'-space and
K a countable r -space. Assume that the stabilizer subgroups ra are exact,
for all a E K, and that there exists a net of Borel maps (n: X ---> Prob(K)
(i.e., the function X 3 x H (~(a) E IR is Borel for every ·a EK) such that

lim r lls.(~ - (~·x11dm(x)=0
n Jx
for every s E r and every regular Borel probability measure m on X. Then
r is exact.

Moreover, if X is amenable as a ra -space for every a E K, then X is

an amenable r -space.