7.3. Homotopy invariance 247Exercise 7.2.4. Use the spectral theorem to show that {N} C IIB(H) is a
quasidiagonal set, for every normal operator N. Hence C*(N) + JK(H) is as
well.Exercise 7.2.5. Let AC IIB(H) be a concrete C*-algebra. Show that A is a
quasidiagonal set if and only if the extension
0 __, JK(H)--> A+ JK(H)--> A/(A n JK(H))--> O
is quasidiagonal (in the sense of Exercise 7.1.5).
Exercise 7.2.6. Use the representation theorem to give a simple proof of
Proposition 7.1.12 (in the unital separable case).Exercise 7.2.7. Formulate and prove a nonunital version of the represen-
tation theorem.
7.3. Homotopy invariance
Quasidiagonality is not particularly well understood. This is probably due
to its topological nature - a point of view first suggested by Voiculescu.
In this section we prove an important theorem which both illustrates this
topological nature and gives us some surprising examples of QD C* -algebras.
We need two quasicentral-approximate-unit facts.
Lemma 7.3.1. Let J <I A be a separable ideal. Then there exists a quasi-
central approximate unit { ej} C J such that ej+l ej = ej for all j E N.Proof. Since J is separable, it contains a strictly positive element h (i.e.,
cp( h) > 0 for all states cp on J -let h = I: lj ej where { ej} is any approximate
unit). For each n let fn E Co(O, 1] be the function which is zero on the
interval (0, Jn], one on the interval [ 2 n^1 _ 1 , 1] and linear in between. Evidently
we have fn+i(h)fn(h) = fn(h), for all n, and it is also readily seen that
cp(fn(h)) --> 1 for all states cp on J. The usual Hahn-Banach convexity
argument allows us to extract a quasicentral approximate unit from the
convex hull of {fn(h)}, so we leave the remaining details to you. D
Lemma 7.3.2. Let E > 0 and a continuous function f E Co(O, 1] be given.
There exists a o > 0 such that for every C* -algebra A and pair of elements
e, a E A in the unit ball of A, withe~ 0, we have
ll[e,aJll < O ==?-jj[f(e),aJll < E.Proof. By a standard approximation argument we may assume that f is a
polynomial of the form
f(s) =tis+ t2s^2 + · · · + tnsn.