8.1. Stable uniqueness 263To get a feel for the type of thing we're after, here's a very important
first step.
Proposition 8.1.3. Assume A is RFD and that it has the following "sta-
ble uniqueness" property: For each pair of *-homomorphisms 0-1, 0-2: A -+
Mk(<C) there exists an integer N such that (N + l)o-1 is unitarily equivalent
to 0-2 EB N 0-1. Then A is AF embeddable.
Proof. Fix a sequence of *-homomorphisms Pi: A -+ Mk(i) (<C) with the
property that
llall = i---+oo _lim llPi(a) II
for all a E A. (Hint: Take direct sums of a separating sequence of rep-
resentations.) Our goal is to use the assumed stable uniqueness property
to construct finite-dimensional C-algebras Bi, injective -homomorphisms
'Tri: Bi -+ Bi+l and *-homomorphisms cri: A -+ Bi such that each CTi con-
tains Pi as a direct summand (which ensures that the CTi 's are asymptotically
isometric) and
CTi+l = 'Trio CTi
for all i. This is evidently more than enough to invoke the previous propo-
sition.
The construction is recursive so let's define Bi = Mk(l) (<C) and 0-1 =
Pl : A -+ Bi. Now consider Bi® Mk(Z) ( <C) and the pair of *-homomorphisms
0-1 ® lk(2) = k(2)cr1: A -+ Bi ® Mk(2) (<C)and
1B 1 ® pz: A-+ B1 ® Mk(2) (<C).
Since these maps take values in the same matrix algebra, we can apply stable
uniqueness to find an integer Ni such that
(N1 + l)k(2)cr1: A-+ Bi® Mk(2)(Ni+l)(<C)
and
(1B 1 ® pz) EB Nik(2)cr1: A-+ Bi® Mk(2)(Ni+i)(<C)
are unitarily equivalent. So let u E Bi® Mk(Z)(Ni+l)(<C) be a unitary such
that
u(N1 + l)k(2)cr1(a)u* = (1B 1 ® pz(a)) EB Nik(2)cr1(a)
for all a EA.
We are ready for the next step: Let Bz =Bi ®Mk(2)(Ni+i)(<C), 7r1: Bi-+
Bz be defined by
7r1 (T) = u(T ® lk(2)(N 1 +1) )u*
and 0-2 : A -+ Bz be given by
cr2(a) = (1B 1 ® pz(a)) EB Nik(2)cr1(a).