1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

274 8. AF Embeddability

hence we can apply Lemma 8.3.4 to C(q) and get a (7r(q)(Jf^1 ), 81)-suitable

u.c.p. map B1(q): C(q)-+ Dl(q) and integers no,q(P) such that

n ( ) ('rr(q)(;yo),4eo)

01 q ~ EB no,q(p)Bo,q(p),

pEPo,p?:_q

where Bo,q(P) = Bo(p)o?T(p)oiJ(q). Thus we can use the integers no,q(p) to get

a connecting map Bo-+ Dl(q) which almost intertwines e7o and B1(q) o?T(q)

(up to 4Eo on Jo) by simply cutting off the summands of Bo which correspond

to those points in Po which are strictly less than q.

Define

and
e71 = ffi fJ1(q): C-+ Bl,
qEP1
where fJ1 (q) = B1 (q) o?T(q). Since we know how to get a map from Bo to each
summand in Bl (which almost intertwines e70 and e71), we can take a direct
sum to get the desired embedding Bo <--+ Bl (which almost intertwines e7o
and e71, as well). In other words, aside from the notational horrors, it is easy
to see that we have constructed a diagram

Bo Bl

'\-- of

O"Q " " / / O"l

" /

c

which 4Eo-commutes on Jo.

Now repeat.... D

8.4. Homotopy invariance

This section establishes the analogue of Voiculescu's homotopy invariance
theorem (Theorem 7.3.6) for AF embeddability of exact C* -algebras. The
proof requires Theorem 8.3.5 plus a number of other facts. The majority
of the extra ingredients are elementary and complete proofs will be given.
However, we need a nontrivial generalization of Voiculescu's Theorem, due
to Kasparov, and refer the reader to [97] for a proof.

Definition 8.4.1. An ideal I <l E is essential if it has "no orthogonal com-
plement" - i.e.,

11-:= {e EE: ex= xe = O, for all x EI}= {O}.

It is a simple exercise to check that an ideal is essential if and only if
every other nonzero ideal intersects it nontrivially.