7.2. The representation theorem 245Proof. With the help of the previous lemma we'll show that Definition 7.2.1
implies a stronger statement: For each finite set ~ C n, x c 1-{ and c; > 0
there exists a finite-rank projection PE IIB(H) such that llPT-TPll < c; for
all TE~ and (here's the point) P(v) = v for all v EX· Once established,
the remainder of the proof is routine since this stronger property evidently
allows one to construct the desired increasing projections.
So, let ~ c n, x c 1-{ and E > 0 be given. If Q is the orthogonal
projection onto the span of x, then, by norm-compactness of the unit ball of
QH, we can find a larger finite set x C QH with the property that llFQ -
Qll < 35, whenever Pis a finite-rank projection satisfying llF(v) - vii < 5
for all v E X· Since D is a quasidiagonal set, we can find such a projection
P which also approximately commutes with ~. Then, if 5 is small, we can
apply the previous lemma to find a unitary U E IIB(H) such that Q :::; U FU
and llU - lll :::; 305; hence, llF - UFUll :::; 605. Defining P = UFU*,
we have constructed a projection which dominates Q and almost commutes
with~ (if 5 is small enough) since it is near P and P almost commutes with
~. D
Definition 7.2.4. Let 7f: A --* IIB(H) be a *-homomorphism. Then 7f is
called a quasidiagonal representation if 7r(A) is a quasidiagonal set of oper-
ators.7Historically quasidiagonality for C* -algebras was defined in terms of the
existence of a faithful quasidiagonal representation. The following result of
Voiculescu - which is the representation theorem referred to. in the title of
this section - shows that the two definitions agree in the separable unital
case.^8
Theorem 7.2.5 (Voiculescu). For a separable unital C*-algebra A, the fol-
lowing statements are equivalent:
(1) A is QD;
(2) A has a faithful quasidiagonal representation (on a separable Hilbert
space);
(3) every faithful unital essential representation of A (on a separable
Hilbert space) is quasidiagonal.Proof. (1) =?-(3): Let 'Pn: A-- Mk(n)(C) be u.c.p. maps which are asymp-
totically multiplicative and isometric. Let 7f: A -- IIB(H) be a faithful es-
sential representation. We must show that if finite sets ~ C A, X C 1i and
7warning: This is not equivalent to saying that 7r(A) is a QD 0*-algebra! Sad, but true (see
Remark 7.5.3).
8They agree in full generality, but this case is sufficient for our purposes.