1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

358 12. Approximation Properties for Groups

s EA) and the unit. We have to show that the set {xs Er: wr(xs) s N}

is finite for every N E N. For this, we claim that

wr(xs) s N ==? {s} U supp(x) c {t EA: WA(t) s N} =: BA(N).

Since BA(N) and Y are finite by assumption, this suffices. Let xs E r

and suppose by contrapositive that there is t E {s} U supp(x) such that

WA(t) > N. Then there are WA(t) many half spaces HE 1-l such that e EH

and t E He, which implies e E E(H, lw) and xs E E(H, lHc )c. (Here lHc

is the unit function.) It follows that wr(xs) > N, so we're done. D

We now introduce the notion of a co-amenable subgroup - a conve-

nient generalization of finite-index subgroups and normal subgroups with

amenable quotients.

Definition 12.2.12. Let A be a subgroup of r. We say that A is co-

amenable in r if there exists a left r-invariant mean μ on .e^00 (r /A).

It can be shown that a subgroup A is co-amenable in r if and only if it

satisfies a co-F¢lner condition: For any finite subset E c r and any c: > 0,

there exists a finite subset F c r /A such that

JsFL.FJ

1.!1E8: JF I < c.

The proof of this fact is a verbatim translation of the proof of Theorem 2.6.8.

Proposition 12.2.13. Let A be a co-amenable subgroup of r. If A has the

Haagerup property, then r has the Haagerup property.

Proof. Let ( 7/Ji) be a net of positive definite functions on A satisfying the
conditions in Definition 12.2.1. We denote the induced positive definite
functions by

,(/Ji: r 3 s ~ L 7/Ji(cr(sx)-^1 scr(x))esx,x E JE(f^2 (r/A)),

xEI'/A

where (}": r I A ----+ r is a fixed cross section. (See Lemma D.2.) For a finite
subset F c r /A, let XF be the characteristic function on F and set

for s Er. It follows that ,(/;i,F is positive definite and, for any c: > 0, the set

{s EI': J,(/;i,F(s)J > c:} C cr(F){t EA: J'l/Ji(t)J > c:}cr(F)-^1

is finite. Moreover, limj limi ,(/;i,Fj = 1 pointwise for a F0lner net ( Fj )j. D