8.5. A survey 279Proof. This follows immediately from Remark 8.4.12 and Proposition 8.4.9,
since Theorem 8.3.5 ensures that CA is AF embeddable. D
Theorem 8.4.14. If A and Bare separable exact homotopy equivalent (Def-
inition 7.3.4) and B is AF embeddable, then A is also AF embeddable.Proof. As with quasidiagonality, it suffices to know that A is homotopi-
cally dominated by B. So assume <p : A -+ B and '!/; : B -+ A are -
homomorphisms and 1t: A -+ A is a path of -homomorphisms such that
'Yo = idA and 'Yl = '!/; o c.p.
By the previous lemma, the mapping cylinder of'!/; EB idB: B -+ A EBB is
AF embeddable and hence we only need to embed A into Z'!/iEBids. For each
a E A let fa: [O, 1] -+ A EB B be defined by
f a(t) = 1t( a) EB c.p(a).The mapping
A-+ ( C[O, 1] Q9 (A EBB)) EBB, a c---t fa EB c.p(a)is easily seen to give the desired *-monomorphism A '---t Z'!/iEBids. D
8.5. A survey
This section is a survey of results and problems; there are essentially no
proofs, just definitions, statements and references.
Recall that a C* -algebra is approximately subhomogeneous (ASH) if it is
an inductive limit of subhomogeneous algebras (Definition 2.7.6).
Proposition 8.5.1. Every ASH algebra is AF embeddable.
Proof. Assume A = LJ An where each An is subhomogeneous. Let B C A*
be the norm closure of the union of the von Neumann algebras A~ CA*.
Each A~ is a (typically nonseparable) AF algebra (being isomorphic to a
finite direct sum of algebras of the form L^00 (X) Q9 Mn(C)) and hence so is
B. Evidently A c B, so we are done. D
In [169] Mikael R¢rdam gave an alternate proof of Theorem 8.3.5 which
is worth outlining. The main result, which is of independent interest, is this:
Theorem 8.5.2 (R¢rdam). There exists an ASH algebra B with the prop-
erty that
3 Actually, B is a nonunital AH algebra and the construction is due to Mortensen. What
Rprdam shows is the tensorial absorption property.