1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.1. Stable uniqueness 265

Lemma 8.1.6. Let ao, a1: A ---+ Mk(C) be homotopic *-homomorphisms

and 1f: A ---+ JIB('H) be any faithful essential representation. Then for each

finite set 'J C A and E > 0 there exists a unitary

U E Clg~EB1i + JK(C~ EB 'H)

such that

llU(ao(a) EB 7r(a))U* - a1(a) EB 7r(a)ll < E

for all a E 'J.

Proof. Note that both ao EB 1f and a1 EB 1f are faithful essential representa-

tions. So the point is not that we can conjugate one to the other -that's just

Voiculescu's Theorem - but rather the placement of the unitary U. Note

also that the Clg~EB1i+lK(£~EB'H) C JIB(C~EB'H) is invariant under conjugation

by unitaries.

Here is the proof:

ao EB 1f rv ao EB (p EB 1f) = ( ao EB p) EB 1f rv ( a1 EB p) EB 1f = a1 EB (p EB 1f) rv a1 EB 1f.

Perhaps a few more details are in order? By the previous lemma we can

find an integer N, a *-homomorphism p: A ---+ MN(C) and a unitary u E

JIB(£~ EB £'jy) such that

llu(ao(a) EB p(a))u* - a1(a) EB p(a)ll < E

for all a E 'J.

By Voiculescu's Theorem (applied to pEB7r and 7r) we can find a unitary

v: 'H ---+ £'jy EB 'H such that

llv1f(a)v* - p(a) EB 7r(a)ll < E

for all a E 'J.

Now define U = (lk EB v)*(u EB lH)(lk EB v). Since the unitary u EB 17-i is

a compact perturbation of the identity, it follows that

U E Clg~EB1i + JK(C~ EB 'H)

and a simple calculation shows that the desired estimate holds. D

We are almost ready for the right stable uniqueness result. We just need
a definition which brings exactness into the picture.

Definition 8.1. 7. Let 'J C A, a finite set, and E > 0 be given. We say
that a finite-dimensional representation a: A ---+ Mk(C) is ('J, s)-admissible
if there exists a faithful essential representation 1f: A---+ JIB('H) such that

(.~,c)

1f Rj (Joo

where a 00 =EB~ a: A---+ JIB(EB~ £~)is the infinite amplification of a.