7.5. External approximation 257Exercise 7.2.4). However, A is evidently isomorphic to C(S EB S EB S) c
B(£^2 (N) EB £^2 (N) EB £^2 (N)). The resulting representation of A can't be quasi-
diagonal since Ind(S* EB S EB S) = -1.
When we further assume that A is exact, a much stronger approximation
is possible -by finite-dimensional algebras.11 Unfortunately, the proof is not
so easy. We will need a technical dilation lemma which is directly inspired
by years of hard work in Elliott's classification program. Indeed, several
variations of the next result can be found in the classification literature where
they are used to prove "uniqueness" theorems for morphisms of various
kinds.
Lemma 7.5.5. Assume that A is unital exact and QD. For every finite
set J C A and c: > 0 there is an essential faithful nonunital representation
7r: A -----+ B(JC), a c.p. map ~: A -----+ B(JC) such that 7r(l) and ~(1) are or-
thogonal projections with lx = 7r(1) + ~.(1); and a unitalfinite-dimensional
subalgebra BC JIB(JC) together with a u.c.p. map 1= A-----+ B such that
lh(a) - (7r(a) EB ~(a))ll < c:
and
lh(ab) - 'Y(a)'Y(b)ll < E
for all a, b E J.
Proof. Assume that A c B(Ji) is a faithful essential representation. Fix a
finite set J c A from the unit ball of A. Enlarging, if necessary, we may
assume that J is self-adjoint and contains the unit of A. Define a larger
finite set
j' ={ab: a,b E J}.
Note that Jc j'.
Since A is QD, the representation theorem (Corollary 7.2.6) provides
us with increasing finite-rank projections Pn ::; Pn+l which asymptotically
commute (in norm) with A and converge to the identity in the strong oper-
ator topology. Define 'Pn: A-----+ PnB(Ji)Pn ~ Mk(n)(C) by 'Pn(a) = PnaPn·
By exactness (Exercise 3.9.5) we can find n large enough that there exists a
u.c.p. map 1/Jn: Mk(n)(C)-----+ JIB(Ji) such that
Ila -1/Jn('Pn(a))ll < c
for all a E j'. We may also assume that
lllt'n(ab) - 'Pn(a)cpn(b)ll < c
llThough the approximants are no longer compact perturbations; they can't be, as this
would imply AF.