5.5. Coarse metric spaces 197
finite direct sum of the nuclear C* -algebras of the form £^00 ® Mk ( <C). For
each x EX, we define Vx E IIB(£^2 (X), £^2 (Bs(x))) by VxOy = (y(x)Oy for every
y E x. It is not hard to check that I:xEX v; Vx = 1 in the strong operator
topology. Hence, we get a u.c.p. map 'lj;: ITxEX IIB(£^2 (Bs(x))) -----* IIB(£^2 (X))
by defining
'lj;((bx)xEX) = L v;bxVx,
xEX
where convergence is in the strong operator topology. Since the support of
v;bVx is contained in T23(X) for any x E X and b E IIB(£^2 (Bs(x))), the
range of 'lj; is actually contained in A(X). Define a positive definite kernel
by k(x, y) = ((y, (x) and denote by mk the corresponding Schur multiplier
- i.e., mk([ax,y]x,yEX) = [k(x, y)ax,y]x,yEX (cf. Appendix D). Then, for any
a E C~(X), one has
('!j; o cp(a)oy, Ox) = L (Vz*aVzOy, Ox)
zEX
= (aoy, Ox) L (y(z)(x(z) = (mk(a)Oy, Ox)
z
and hence 'lj; o cp = mk. Therefore, for a E i, we have
11('1/J o cp)(a) - ail :S llall sup 11-k(x, y)I sup IBR(x)I < c:.
(x,y)ETR(X) x
This proves the nuclearity of C~(X).
Now assume that C~(X) is nuclear and let R > 0 and c: > 0 be given.
Since X has bounded geometry, one can find a finite set i of partial isome-
tries in A(X) with the property that for every (x, y) E TR(X), there exists
v E i such that vOx == Oy. Since C~(X) is nuclear, there exist u.c.p. maps
cp: C~(X) -----* Mn(<C) and 'lj;: Mn(<C) -----* C~(X) such that II ('l/; ocp)(v)-vll < c:
for every v E i. We set 1i = .e; @£;and consider the Hilbert C~(X)-module ·
1i ® C~(X). (For the rest of this proof, inner products are linear in the sec-
ond variable.) Note that Mn(q ®Mn(<C) naturally acts on 1i®C~(X) from
the left. Let {eij} be matrix units for Mn(<C) and observe that ['l/;(eij)] is
positive in Mn(C~(X)), by complete positivity of 'lj; (see Proposition 1.5.12).
Let [bij] = ['l/;(eij)]^112 E Mn(C~(X)) and
·e'lj; = L:(j ® (k ® bkj EH® c~(x), -
j,k
where (j denotes the standard basis of .e;. It is routine to check that '!j;(a) =
(e'!f;, (a® l)e'!f;) for every _a E Mn(<C). Perturbing e'!f;, we .may assume that
e'!/J E 1i ® A(X). We write e'!/J = l:z 6 ® az and set
(x(z) = II L 6(0z, azOx)e2(x) llH
l
