7.5. External approximation 259It will be convenient to introduce some notation. Let i.p: A--+ IIB(H) and
'ljJ: A--+ IIB(JC) be u.c.p. maps and suppose that a finite set~ c A and c > 0
are given. Then we writeif there is a unitary operator U: H--+ JC such that 111.p(a) - U*'ljJ(a)Ull < c
(~,10) (~,8) (~,10+8)
for all a E ~- Note that if <p ~ 'ljJ and 'ljJ ~ ry, then <p ~ ry.
Theorem 7.5.7 (Dadarlat). Let A be an exact QD C*-algebra and 1f: A--+
IIB(H) be a faithful essential representation. For each finite set ~ c A and
c > 0, there exists a finite-dimensional C* -algebra B C IIB(H) such that for
each a E ~ there exists b E B with Ila - bll < E.Proof. We may assume A is unital and that a finite set ~ C A and c > 0 are
given. Since all faithful essential representations are approximately unitarily
equivalent, it suffices to show that some representation has the property
stated in the theorem. By the previous lemma we can find an essential
faithful nonunital representation 1f: A --+ JB(JC), a c.p. map : A --+ IIB(JC)
such that 1f(l) and (l) are orthogonal projections with lK = 1f(l) + (l),
and a unital finite-dimensional subalgebra B C IIB(JC) together with a u.c.p.
map ry: A --+ B such that
lll'(a) - (7r(a) EB (a))ll < c
and
llf'(ab) - ry(a)ry(b)ll < c
for all a, b E ~-We now apply Arveson's oo + 1 = oo trickery:
(~,10)
/' ~ 1f EB <I>
(~,10)
~ (1fEB1f)EB<I>
= 1f EB (1f EB <I>)
(~,e)
~ 1fEB"(
(~,2fi)
~ 1r.If you believe each step above, then the proof is complete since
(~,3e+2fi)
'Y ~ 1f
and the range of ry is contained in a finite-dimensional C*-algebra. So let's
justify each step (after the first, which is immediate). The second line follows
from Voiculescu's Theorem since 1f and 1f EB 1f are both faithful, essential
representations. The third is trivial while the fourth line follows from the