5.1. Exact groups 169Remark 5.1.5. Note that if k is a positive definite kernel supported in a
tube (i.e., k(s, t) = 0 for all (s, t) outside some tube), then we can identify
k with an element in C~(r), since [k(s, t)]s,tEr obviously defines a bounded
operator on £^2 (r). Since an operator is positive if and only if all compressions
by finite-rank projections are positive matrices, we see that there is a one-
to-one correspondence between positive definite kernels supported in tubes
and positive operators in the *-algebra generated by >.(qr]) and f^00 (r).The following is largely a translation of the results from Section 4.4. The
only new addition is (1) ==?-(2).
Theorem 5.1.6. Let r be a discrete group. The following are equivalent:
(1) r is exact;
(2) for any finite subset E c r ands > 0, there exists a positive definite
kernel k: r x r -+ c whose support is contained in a tube and such
that
sup{lk(s, t) - ll : (s, t) E Tube(E)} < s;
(3) for any finite subset E c r and s > 0, there exist a finite subset
F c rand(: r-+ f^2 (r) such that ll(tll = 1, supp(t c Ft for every
t Er and
sup{ II Cs - (tll : (s, t) E Tube(E)} < s;
(4) for any finite subset E c r ands > 0, there exist a finite subset
F c r and μ: r -+ Prob(r) such that supp μt c Ft for every t E r
and
sup{\\μs - μtll : (s, t) E Tube(E)} < s;
(5) c~ (r) is nuclear.Proof. (1) ==?-(2): Let a finite subset E c r ands > 0 be given. It follows
from Exercise 3.9.5 that we can find a finite subset F c r with the following
property: If 1.p: llll(f^2 (r))-+ llll(f^2 (F)) is compression by the projection onto
R,^2 (F) c £^2 (r), then there is a u.c.p. map 'ljJ: llll(f^2 (F))-+ llll(f^2 (r)) such that,
setting e = 'ljJ 0 i.p' one has
max llB(>.(s)) - >.(s)ll < s.
sEE
We define a kernel k: r x r -+ C by
k(s, t) = (B(>.(sc^1 ))8t, Os)·
This kernel is positive definite. Indeed, for all s1, ... , Sn E r and a1, ... , an E
C, one has
Lk(si,sj)aiaj = L(e(>.(sisj^1 ))ajOsj,aiosJ
i,j i,j