278 8. AF Embeddability
Proposition 8.4.9. Assume 0--* I--* D --* B--* 0 is exact and there is a
*-homomorphic splitting (}: B --* D. If I <ID is essential and both I and B
are AF embeddable, then D is also AF embeddable.Proof. By Lemma 8.4.5 we may assume that I is actually an AF alge-
bra. (The larger exact sequence obviously splits since the smaller one
does.) Tensoring everything with IK, if necessary, we may assume I ~
IK@I. By Corollary 8.4.7, there is an embedding of D into the C*-algebra
7r(B)@ Cl+ IK(JC)@ I, where Jr: B--* JB(JC) is any faithful essential repre-
sentation. But, since B embeds into some AF algebra C, we can take 7r to
be a representation of C and then Lemma 8.4.8 ensures that
(7r(C) + IK(JC))@ (Cl+ I)
is an AF algebra which contains D. DAfter a few more remarks, we can harvest the homotopy invariance the-
orem.Definition 8.4.10. If 'lj;: B --* A is a *-homomorphism, then the mapping
cylinder of 'lj; is the C* -algebra
Z.p := {f EB b E (C[O, l] Q9 A) EBB: f(l) = 'lj;(b)}.
Remark 8.4.11. We have a surjective coordinate mapping Z.p --* B and
evidently the kernel of this map can be identified with CA = C[O, 1) @ A -
i.e., we have a natural short exact sequence
0 --* CA--* Z.p --* B --* 0
and this sequence has a *-homomorphic splitting defined by b i-. 'lj;(b) EB b
(where we think of the left side as being a constant function).
Remark 8.4.12. Note that in the case 'lj; is injective, the other coordinate
mapping Z.p--* C[O, l]@ A is an isomorphism onto the algebra
{f E C[O, l] @A: f(l) E 'lj;(B)}.It is easily seen that CA will be an essential ideal in Z.p in this case - for
any nonzero path we can construct an element of CA which won't multiply
to zero - and hence the sequence
0--* CA--* Z.p --* B--* 0is essential with a *-homomorphic splitting.
Lemma 8.4.13. Assume 'lj;: B -- A is an injective -homomorphism such
that B is AF embeddable and A is exact. Then the mapping cylinder Z.p is
also AF embeddable.