196 5. Exact Groups and Related Topics
(1) X has property A;
(2) for any R > 0 and E > 0, there exists a positive definite kernel k
on X such that k(x, x) = 1 for every x E X, ll - k(x, y)I < E if
(x, y) E TR(X), and supp k c Ts(X) for some S > O;
(3) for any R > 0 and E > 0, there exist a map (: X ___,. £^2 (X) and
S > 0 such that JJ(xll = 1 for all x, il(x-(yjj < E for (x, y) E TR(X) 1
and supp(x C Bs(x) for every x.Proof. (1) '* (2): Set k(x,y) = (~y,~x)·
(2) ' (3): Since k has finite propagation, we may regard k as a positive
bounded operator ak on £^2 (X) which belongs to the translation algebra
A(X). Since the operator aif^2 is in C~(X), the closure of A(X), there is an
operator b E A(X) such that llak - bbll < E. Let 'f/x = Mx, where { 8x}xEX
is the canonical orthonormal basis of £^2 ( X). Since b has finite propagation,
there exists S > 0 such that SUPP'T/x C Bs(x) for every x. Moreover, we
have
('Tfy, 'Tfx) = (b*My, 8x) ~,, (ak8y, 8x) = k(x, y).
Hence, defining (: X ___,. £^2 (X) by (x = ll'Tfxl!-^1 'Tfx, we see that ((y, (x) ~
(ryy, 'T/x) and thus (satisfies condition (3), modulo a change of E (since ll'Tfxl! ~
1, for all x EX).
(3) '* (1) is obvious. D
Theorem 5.5.7. Let r be a countable discrete group. Then r has property
A if and only if r is exact. More generally, for a discrete metric space X
with bounded geometry, property A is equivalent to the uniform Roe algebra
C~(X) being nuclear.
Proof. The first part follows from the previous lemma and Theorem 5.1.6.
For the second part, first suppose that X has property A. To show
C~ ( X) is nuclear, it suffices to show that. the identity map on C~ ( X) is
approximated by u.c.p. maps which factor through nuclear C* -algebras. Let
a finite subset i C A(X) and E > 0 be given. Choose some large R > 0
such that the support of a is contained in TR(X) for every a E i. By
assumption, there exist (: X ---i· £^2 (X) and S > 0 satisfying condition (3)
in Lemma 5.5.6 for R > 0 and (maxaE~ l!al! supx JBR(x)l)-^1 E. We define a
u.c.p. map cp: C~(X) ___,. llxExlIB(£^2 (Bs(x))) by
cp(a) = (cpx(a))xEX,
where cpx: lIB(£^2 (X)) ___,. JIB(£^2 (Bs(x))) is the standard compression map. Re-
call that there is an identification llnENMk(C) ~ £^00 (N) 181 Mk(C). Since
X has bounded geometry, there is a uniform bound on the dimensions of
the Hilbert spaces £^2 (Bs(x)); thus llxEXJIB(£^2 (Bs(x))) is nuclear, being a