1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

7.1. The deE.nition, easy examples and obstructions 239

Another thing one should check is whether quasidiagonality passes to

unitizations. It does (Exercise 7.1.2); thus most quasidiagonal questions

can be reduced to the unital separable case.

How about some simple examples and permanence properties?

Proposition 7.1.5. Every abelian C*-algebra is QD.

Proof. Use direct sums of point evaluations to construct finite-dimensional

*-homomorphisms. D

Abelian algebras are particular examples of residually finite-dimensional

C* -algebras. Many QD questions can be reduced to these natural analogues

of block diagonal operators on a Hilbert space.

Definition 7.1.6. A C*-algebra A is called residually finite-dimensional

(RFD) if there exist finite-dimensionah-homomorphisms 1T n : A ---t Mk( n) ( q

such that ffi?Tn: A---t f1Mk(n)(C) is faithful.^2

The following fact should be obvious.

Proposition 7.1.7. Residually finite-dimensional C*-algebras are QD.

Here is an often forgotten class of RFD C* -algebras.

Proposition 7.1.8. Every type I C*-algebra with a faithful tracial state is

RFD (hence QD).

Proof. Let T be a faithful trace on a type I algebra A and let ?T 7 : A ---t
1B(L^2 (A, T)) be the associated GNS representation. Since Tis faithful, so is
1T 7".
By the structure theory of type Ivon Neumann algebras we have
1T 7 (A)" '.2: IT £CX'(Xn, μn) ® JB(Hn)·
n
But ?T 7 (A)" has a faithful trace (the vector state in the GNS representation
is tracial and faithful) and hence the dimensions of all the Hn's must be
finite. Since it is simple to show that C(X) ®Mk(C) is RFD, the remainder
of the proof is straightforward. D

Of course, finite-dimensional algebras are QD, so the next proposition

implies the same for AF algebras. In fact, since subhomogeneous algebras

(Definition 2.7.6) are RFD - the direct sum of all irreducible representa-

tions is always faithful - it follows that inductive limits of subhomogeneous

algebras (aka ASH algebras) are also QD.

2In the nonseparable setting one may need uncountably many representations, of course.