1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

170 5. Exact Groups and Related Topics

2: 0

since e is c.p. and [.A(sisj^1 )]i,j 2: 0 in Mn(C~(r)) (Example 1.5.13). Since

<p(.A(s)) = 0 when sF n F = 0, the support of the kernel k is contained in

Tube(F p-l ). Moreover, if (s, t) E Tube(E), then

k(s,t) = (O(.A(sr^1 ))ot,Os) ~c (.A(sC^1 )0t,Os) = 1.

(2) ::::} (3): Let a finite subset E c r and E > 0 be given and take a

kernel k satisfying condition (2). Denote by ak E C~(r) the positive operator

corresponding to the kernel k (see Remark 5.1.5). Since ai^12 E C~(r), there

exists b E IIB(.e^2 (r)) with llak-b*bll < E such that suppb c Tube(F) for some

finite subset F c r. Set T/t =Mt E .e^2 (r). We note that llTJtll^2 ~c (akOt, Ot) =

k(t, t). It follows that supp'Tft c Ft for every t Er and

(TJt, T/s) = (b*Mt, Os) ~c (akOt, Os)= k(s, t).

Hence, (t = TJt/llTJtll satisfies condition (3) (with E modified).

(3) <=? (4): Observe that the map .e^2 (r) 3 ( f----+ 1(1^2 E .e^1 (r) is uniformly

continuous on the unit sphere.

( 4) ::::} (5): Thanks to Proposition 5.1.3 and Theorem 4.4.3, it suffices

to show the translation action of r on .e^00 (r) is amenable. So, fix a finite

symmetric set E c r and E > 0. Choose some μ: r ----t Prob(r) as in

condition ( 4).

Define a function T: r----+ .e^00 (r) by T(g)(x) = Jμx(x-Ig). As in the

proof of Lemma 4.3.7, one checks that for every s Er,

lls *a T-Tll~ =sup (2: IV μs-^1 x(x-1g) _ J μx(x-lg)/2)

xEI' gEI'

-1

:S sup llμs x - μxiii·

xEI'

However, (s-^1 x,x) E Tube(E) whenever s-^1 E E and hence we see that
Ifs *a T -Tll2 is small for alls EE. The proof of Lemma 4.3.7 shows that
we can replace T with a function of finite support, and this completes the
proof.

(5) ::::} (1): Trivial. D

Here is a purely dynamical characterization of exactness.

Theorem 5.1. 7. For a discrete group r, the following are equivalent:

(1) r is exact;