16.2. C'-preliminaries 42316.2. C*-preliminariesTo solve Herrero's problem, we will need a technical generalization of Dadar-
lat 's approximation theorem for exact QD C*-algebras (compare Theorem
7.5.7 with Theorem 16.2.6 below). This, in turn, requires a few preliminary
results.
Lemma 16.2.1. If A C IIB(H) is a unital exact C* -algebra, then A+ OC(H)
is also exact.
Proof. Since exactness passes to quotients (Corollary 9.4.3) and locally split
extensions of exact C*-algebras are again exact (Exercise 3.9.8), it suffices
to show that the extension
0----; OC(H)----; A+ OC(H)----; A/(A n OC(H))----; O
is locally split. Since quotient mappings of exact C* -algebras are always
locally liftable (Remark 9.4.2), the related extension
0 ----; An OC(H) ----; A ----; A/ (An OC(H)) ----; 0
is locally split. Evidently this implies what we want. D
The following corollary isn't really necessary, as all the arguments which
follow can be localized to finite-dimensional operator subsystems. However,
it makes things a little cleaner (and follows immediately from the previous
fact and Remark C.5).
Corollary 16.2.2. If A C IIB(H) is unital and exact, then there exists a
u.c.p. splitting for the sequence
O ----; OC(H) ----; A+ OC(H) ----; A/ (An OC(H)) ----; 0.
Here is another requisite application of local liftability.Proposition 16.2.3. If A C IIB(H) is a quasidiagonal set of operators and A
is exact, then 7r(A) is exact and QD, where 7f: IIB(H) ----; Q(H) is the quotient
map to the Calkin algebra.
Proof. We may assume A is unital. Since A is a quasidiagonal set of oper-
ators, the extension
O----; OC(H)----; A+ OC(H)----; A/(A n OC(H))----; 0
is quasidiagonal (Exercise 7.2.5). Since A+ OC(H) is exact, the quotient
mapping A+ OC(H) ----; A/(A n OC(H)) is liftable and hence Exercise 7.1.5
completes the proof.
The reader who didn't do those exercises is probably a bit annoyed right
now. Since we actually need the proofs more than the results, here are the
details.