264 8. AF EmbeddabilityBy construction we have n1 o 0"1 = 0"2.
If you really understand the argument so far, the rest of the proof is
routine. DActually, the previous proposition is completely useless (see Exercise
8.1.1). But 1 the proof is very important. In fact, one should absorb it
completely before proceeding.
Not yet knowing what the right stable uniqueness property is, a clue is
provided by homotopic *-homomorphisms.
Lemma 8.1.4. Let o-o, 0"1: A -+ Mk(C) be homotopic *-homomorphisms
(Definition 7. 3. 3). Then for each finite set :;y c A and s > 0 there exists an
integer N, a *-homomorphism p: A-+ MN(C) and a unitary u E Mk+N(C)
such that for all a E i,
llu(O"o(a) EB p(a))u* - 0"1(a) EB p(a)ll < s.
Or, in the notation prior to Theorem 7. 5. 7,
(;3',e)
O"o EB p ~ 0"1 EB p.Proof. The proof amounts to a simple, but useful, trick. Let O"t: A -+
Mk ( q be a. continuous family of homomorphisms connecting O"Q and 0"1.
Since t f-7 O"t(a) is norm continuous for each a E A, uniform continuity
provides an integer M with the property that
110" i M (a) - O"i+i M (a)ll < s
for all a E :;y and every 0 ::; i::; M -1. One then defines N = k(M -1) and
p = (j h EB (j ..?,__ EB ... EB (j M-1.
M M M
Note that k+N = kM, so we can identify Mk+N(C) with Mk((['.) ®MM((['.).
Finally, let U E MM(C) be the cyclic shift unitary of order Mand define
the unitary we're really after by
u = lk@ U E Mk((['.)@ MM(C) = Mk+N(C).
A straightforward calculation completes the proof. DRemark 8.1.5. Let A be a unital residually finite-dimensional C-algebra
with the property that any two unital -representations on the same Hilbert
space are homotopic (e.g., the unitized cone over an RFD algebra). In this
case one always has the stable uniqueness property above to work with. Try
to mimic the proof of Proposition 8.1.3 and deduce AF embeddability of
such algebras. Don't spend too much time though - it's impossible! The
point here is to identify where the proof of Proposition 8.1.3 breaks down
so we can head toward a better stable uniqueness result.