268 8. AF EmbeddabilitySo, let Br = Mk(I)(<C) and 0"1 = PI: CA -+ Br. Now consider Br®
Mk( 2 )(<C) and the pair of *-homomorphisms
O"I 01k( 2 ) = k(2)0"1: CA-+ Br 0 Mk(2)(<C)
and
1B 1 0 p2: CA-+ Br 0 Mk(2)(<C).Note that 0"1 ® lk( 2 ) is still (~h, .s1)-admissible and homotopic to 1B 1 0 P2·
Hence we can apply Theorem 8.1.8 to find an integer Ni such that
(~1fi ,3ci)
(N1+1)(0"1 ® lk(2)) R:J (1B 1 0 P2) EB N(0"1 0 lk(2))·The remainder of the proof is very similar to that of Proposition 8:1.3;
however one important point should be mentioned: In order to repeat the
recursive procedure, you have to know that (1B 1 0 p2) EB N(0"1 0 lk(2)) is
(i2, .s2)-admissiblef In other words, adding on 0"1 does not decrease the level
of admissibility. Luckily, this is the content of Lemma 8.2.2. D
The following corollary will play a crucial role in showing that exactness
passes to quotients. It is due to Kirchberg, though his proof was quite
different. We say C is a subquotient of A if there exists a subalgebra B c A
with ideal J <l B, such that C = B / J...Corollary 8.2.5. Every separable exact C* -algebra is a subquotient of a
UHF algebra.Proof. Since every algebra is a quotient of its cone, this follows immediately
from Dadarlat's embedding theorem together with Corollary 7.3.8. D8.3. Cones over general exact algebras
We now wish to remove the RFD hypothesis from Dadarlat's result. Though
the main result of this section is a generalization of what we already know,
one of the key steps in the proof is actually more elementary; the right
stable uniqueness result is not very deep. However, setting everything up
is far more complicated and the notation alone will tax one's memory. We
start with an obvious adaptation of the admissible representations which
were so important in the last section.
Definition 8.3.1. Let a finite set i c A c JIB(H) and E > 0 be given. We
say that a u.c.p. map cp: A-+ Mk(<C) is (i,.s)-multiplicative if
max{jjcp(a*a) - cp(a*)cp(a)ll, llcp(aa*) - cp(a)cp(a*)ll} < E.
aE~