1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

266 8. AF Embeddability

Theorem 8.1.8. Let cro, cr1: A--+ Mk((['.) be homotopic *-homomorphisms
and assume that cro is (~, c:)-admissible. Then there exists an integer N such
that

Proof. In a line, here is the proof:

(cro)oo "'cro EB 7r "'cr1EB7r"' cr1 EB (cro)oo·

More precisely, since cro is (~, c}admissible, we can find a faithful essential
representation 7r: A --+ JIB( EBr .e~) such that

ll(cro)oo(a) - 7r(a)ll < c:

for all a E ~- (Just replace whatever faithful essential representation the
definition gives you by a unitarily conjugate representation on the Hilbert
space EBr .e~.) By the previous lemma there exists a unitary
00
U E Cle~EB[EBf' e~J + K(.e~ EB [EB.e~])
1
such that
llU(cro(a) EB 7r(a))U* - cr1(a) EB 7r(a)ll < c:

for all a E ~. Hence the triangle inequality implies

llU(cro(a) EB (cro)oo(a))U* - cr1(a) EB (cro)oo(a)ll < 3c:

for all a E ~.

But why the big deal about our unitary being a compact perturbation of
a scalar? Well, letting PM E IIB(.e~ EB [EBr .e~]) be the orthogonal projection
onto the subspace
M
.e~ EB [ E9 .e~] ,
1
it follows that
ll[Pm, U]ll-+ 0

as M--+ oo. Hence, by standard perturbation theory, we can find unitaries
UM E JIB(.€~ EB [Ef1f1" .€~]) such that llUM - PMUPMll --+ 0 as M --+ oo. It
follows that

limsup llUM((M + l)cro(a))UJkz-- cr1(a) EB Mero( a) II :S 3c:

M-+oo

for all a E ~' so the proof is complete. D

E:xercise

Exercise 8.1.1. Show that the hypotheses of Proposition 8.1.3 imply that
A is a finite-dimensional matrix algebra.