8.3. Cones over general exact algebras 273Go back and forth until the N - 1 step and you havev(O)EBm(l)v(l)EB· · ·EBm(N-l)v(N-1) Ct6,3so) ~ (~ ~ no(i)Bo(i) ) EBμ(N-1),
where μ(N -1) is a map which factors through C(PN-i). Finally, we come
to the end because there is an integer n 0 (N) such that
ct6,so)
μ(N - 1) ~ no(N)Bo(N).
Indeed, Bo(N) factors through C(pN) = C(l) = C; hence we may assume
its range is just C - and μ(N -1) factors through C(PN-i), which means it
maps every element of i'b to a matrix which is within co of a scalar multiple
of the identity. Thus we have
cib' ,4so) }1
v(O) EB m(l)v(l) EB··· EB m(N - l)v(N -1) >:::J wno(i)Bo(i)
i=i
and the proof is complete. D
Theorem 8.3.5. The cone over every separable exact C* -algebra A is AF
embeddable.
Proof. As before C denotes the unitization of CA. It suffices to construct
finite-dimensional C*-algebras Bo, Bi, ... , embeddings Bi<---+ Bi+l and u.c.p.
maps (Ji : C ----t Bi which satisfy the hypotheses of Proposition 8.1.1.
Fix an increasing sequence of finite sets of unitaries io C ii C i2 C · · ·
such that the linear span of their union is dense in C. Define En = 1/2n
and choose some finite sets Pn c [O, 1] which satisfy condition (3) of Lemma
8.3.4 for in and En· Let 6n < En have the property asserted in Lemma 8.3.4
(applied to in, En and Pn)·
To put the machine in motion, we choose any collection of ('rr(p)(ib'^0 ), 60)-
suitable u.c.p. maps Oo(P): C(p) ----t Do(p), where p runs through the points
in Po and the Do(p)'s are matrix algebras. We then let
Bo= EB Do(P)
pEPoand
(Jo= EB Bo(P): C ----t Bo,
pEPo
where Bo(P) = Oo(P) o n(p).
Here is how we get Bi: For each q E Pi we want to construct a
(n(q)(if^1 ),6i)-suitable u.c.p. map Oi(q): C(q) ----t Di(q) and a nice connect-
ing map Bo ----t Di(q) (which won't be injective for all q's but will be for the
smallest q = 0 E Pi). The key remark is that C(q) is isomorphic to C and