7.2. The representation theorem 243Exercise 7.1.3. Show that A is QD if and only if there exists an injective
*-homomorphism
A-+ ITnENMn(CC)
ffinEN Mn(CC)
which admits a c.c.p. splitting A-+ ITnENMn(CC).
Exercise 7.1.4. Let Pi :::; P2 :::; · · · E IIB(7t) be finite-rank projections which
converge strongly to the identity and denote by C the subset ofII PnIIB('lt)Pn
nEN
consisting of those sequences (Tn) for which there exists TE IIB(7t) such that
Tn -+Tin the strong- topology (i.e., Tn-+ T and T:;, -+ T in the strong
operator topology). Prove that C is a C-subalgebra of ITnEN PnIIB(1t)Pn.
Show that IIB(7t) is a quotient of C. Since C is clearly RFD, it follows that
every separable C -algebra is a quotient of an RFD algebra.
Definition 7.1.19. An extension 0-+ J-+ A-+ A/ J-+ 0 is called quasi-
diagonal if J contains a quasicentral approximate unit consisting of projec-
tions.
Exercise 7.1.5. Show that if 0-+ J-+ A-+ A/ J-+ 0 is a quasidiagonal
extension, A is QD, and the extension is locally split, then A/ J is also
QD. (Hint: If {pn} E J is a quasicentral approximate unit of projections
and q?: A/ J -+ A is a u.c.p. splitting, then the nonunital maps q?n(b) =
(1 - Pn)q?(b)(l - Pn) are asymptotically multiplicative. Now localize this
fact.)
Exercise 7.1.6. Show that if 0 -+ J-+ A-+ A/ J-+ 0 is a quasidiagonal
extension and both J and A/ J are QD, then so is A.
7.2. The representation theorem
Though Definition 7.1.1 is perfectly natural from the C-perspective, qua-
sidiagonality was originally imported from single operator theory and for-
mulated in terms of representation theory. The interested reader can find
more on the operator theory origins in Chapter 16 but in this section we
restrict our attention to the C -ideas and prove the fundamental represen-
tation theorem.
Definition 7.2.1. Let n c IIB(7t) be an arbitrary collection of operators.
Then n is called a quasidiagonal set if for each finite set :;y c n, each finite
set x c 1t and each c: > 0 there exists a finite-rank projection P E IIB(7t)
such that
llPT -TPll < E