246 7. Quasidiagonal C* -AlgebrasE > 0 are given, then one can find a finite-rank projection which almost
commutes (up to E) with .;y and almost leaves x fixed.
The trick is simply to apply the technical version of Voiculescu's Theo-
rem (version 1.7.6) to an appropriate faithful *-homomorphism modulo the
compacts. The right thing to consider is the u.c.p. map
00 00 00
<I>: A---+ IT Mk(n)(C) C lIB(E9£~(n)), <I>(a) =EB 'Pn(a).
n=l n=l n=l
As in the previous section, <I> is a faithful representation modulo the com-
pacts. Moreover, if we first fix a finite set .;y c A and E > 0, then (going out
far enough in the sequence) we may assume that each of the u.c.p. maps 'Pn
is almost multiplicative on .;y. In other words, we may assume that rJq,(a) < E
for all a E .;y (notation as in the statement of Theorem 1.7.6). Hence we can
find a unitary U: 1i ---+ EBf' .e~(n) such that ll?r(a) - U*<I>(a)Ull < c: for all
a E i. It should now be clear how to complete the proof since <I> obviously
has finite-rank projections which commute with its image and tend strongly
to one (which we can transfer over to 1i by U* · U).
(3) ::::;,. (2) is immediate so let's show (2) ::::;,. (1). Let 7r: A ---+ JIB(H)
be a faithful quasidiagonal representation and P1 ::; P 2 ::; · · · be finite-
rank projections which converge to the identity strongly and asymptoti-
cally commute with 7r(A). A straightforward bit of estimating shows that
'Pn: A---+ PnlIB(H)Pn ~ Mk(n)(C) defined by 'Pn(a) = Pn7r(a)Pn is an asymp-
totically multiplicative and isometric sequence of u.c. p. maps. DIn most instances, the representation theorem above gets used in the
following form.
Corollary 7.2.6. Assume A is unital separable and QD. For each faithful
essential representation 7r: A ---+ JIB('h'.), with 1i separable, there exist finite-
rank projections P1 ::; P2 ::; · · · , converging strongly to 1 and such that
ll[Pn,?r(a)Jll---+ 0 for all a EA.
Exercises
For the following exercises you should assume that everything in sight
is separable.
Exercise 7.2.1. Show that every representation of a simple unital QD al-
gebra is quasidiagonal.
Exercise 7.2.2. Show that every representation of an AF algebra is quasi-
diagonal.
Exercise 7.2.3. Observe that if Ac JIB(H) is a quasidiagonal set of opera-
tors, then so is A+ JK(H).