1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

208 5. Exact Groups and Related Topics

Thus, writing z = L, >..(fj)<g>aj, we have z*z E Co(G(0))0A C C)'.(G)®maxA.

Since C 0 (GC^0 )) is nuclear, we have

llz*zl/o~(G)®maxA = sup II L L fi('Y)fj(t)a;ajllA ·

xEG(O) i,j "(EGa:

=sup II L fi(t)fj(t)a;ajllA

"(EK i,J ..

=sup II L fj(t)ajllA

"(EK j

as desired.

For a compact subset K c G, we denote by C(K, A) the set of all A-

valued continuous functions on G whose supports are contained in K. It

is clear that C(K, A) is a Banach space under the supremum norm. Let

lK: C(K, A) --+ C)'.(G) ®max A be the natural embedding. By the result of

the previous paragraph, we have lliKll :::; CK, while it is trivial to see that

ll(Q o iK)(f)llo~(G)®A 2:: llflloo for every f E C(K,A). This means that Q is

injective on lK(C(K, A)) for every compact subset KC G.

Now take hi E Cc(G) as in condition (2) and let Ki= supp hi· We may

assume that sup lhi(t)I :::; 1. By Proposition 5.6.16, the multiplier maps

mhi are c.c.p. on C)'.(G) and each mhi ®max idA maps C)'.(G) ®max A into

lKi ( C(Ki, A)) because lKi ( C(Ki, A)) is closed. Since mhi --+ id in the point-

norm topology by Lemma 5.6.12, the net mhi ®max idA converges to the

identity on C>: ( G) ®max A. Let x E ker Q be given. Then we have

Q((mhi ®maxidA)(x)) = (mhi ®idA)(Q(x)) = 0.

This implies (mhi®maxidA)(x) = 0 for alli, so x = lim(mhi®maxidA)(x) = 0.

(3) =? (1): Fix c > 0 and a compact subset KC G. Let U1, ... , Um be

a relatively compact open covering of K such that, for every l, both s and

r are homeomorphic on some neighborhood Vz of Ul. Let fl E Cc(G) be a

function with 0 :::; fz :::; 1 such that fl = 1 on Uz and fz = 0 off Vz. Then

fl>..(fl) II :S 1 and (ft* fz)(x) = 1 for x E s(Ul)· Let 'ljJ: C)'.(G) --+ Mn(C) and

r.p: Mn(C)--+ C)'.(G) be c.c.p. maps such that

ll(r.p^0 1/J)(>..(fl)) - >..(fz)ll < c and ll(r.p^0 1/J)(>..(ft fl)) - >..(ft fz)ll < c

for all l. Let [bi,j] = [r.p(ei,j)]^112 E Mn(C)'.(G)) and set

1]cp = L ~j ® ~k ® bk,j E .e; ® .e; ® C)'. ( G)'

j,k

where ff i }r is the standard basis of .e;. We view .e; ® .e; ® C)'. ( G) as a
Hilbert C)'.(G)-module. Then we have r.p(a) = (1Jcp, (a® 1 ® l)77cp) for every