1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

260 7. Quasidiagonal C* -Algebras

first. Finally the last line also follows from Voiculescu's Theorem (Corollary

1.7.7) since we assumed that 'Y is €-multiplicative on J. D

In view of Remark 7.5.6, the proof above implies the following corollary.

Corollary 7.5.8. Let A c IIB(1i) be an essential representation of an exact

RFD C* -algebra. For each finite set J c A and c > 0 there exist a full

matrix algebra Mk(C) C IIB(1i) and a *-homomorphism TA-+ Mk(C) such

that Ila - 'Y(a)ll < c for all a E J.

Proof. Since A is RFD, we may start with an embedding

AC IJ Mk(n)(C) C IIB(E9£%(n)).

nEN nEN

Now reread the proofs of the previous two re~ults.

Exercise

D

Exercise 7.5,1. Prove the converse of Theorem 7.5.7. That is, if A C IIB(1i)
and every finite subset of A is close to some finite-dimensional subalgebra of
IIB(1i), then A is exact and QD. (This fact was first observed by Voiculescu.
It was a crucial ingredient in the resolution of Herrero's approximation prob-
lem.)

7.6. References

The representation theorem (Theorem 7.2.5) and the homotopy invariance

theorem (Theorem 7.3.6) come from [191]. Choi proved his residual finite-

dimensionality result in [37]. The results of Section 7.5 are taken from [49].