1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Chapter 8

AF Embeddability

Deciding when a particular C* -algebra is isomorphic to a subalgebra of some

AF algebra-i.e., when it is AF embeddable -can be a very challenging prob-

lem. The seminal result in this direction is due to Pimsner and Voiculescu

who used an AF embedding to compute the range of the unique trace on the

irrational rotation algebras. Since then a number of authors have studied

AF embeddability; this chapter introduces some of the techniques used and,

at the end, gives a quick survey of the current state of affairs.

We have no intention of giving a comprehensive treatment, as the proofs

of AF embeddability tend to be quite difficult and different situations require

different techniques. On the other hand, there is one reasonably accessible

case which demonstrates some common themes used in other cases and, most

importantly, has a useful application to exactness, namely, the fact, due to

Dadarlat, that the cone over a separable exact RFD algebra is always AF

embeddable. In Section 8.3 we'll prove a more general result, but the RFD

case is much easier to digest, so we start there.

Throughout this chapter A and B will denote separable C* -algebras.

8.1. Stable uniqueness and asymptotically commuting

diagrams

Let's think about what it takes to embed something in an AF ·algebra.
Lacking an abstract characterization of subalgebras. of AF algebras, there
is re~lly no way to· avoid a construction of some sort. In this section we
introduce some common techniques, starting off gently and progressively
generalizing. In one way or another, most AF-embedding results depend on
the following simple fact.

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