252 7. Quasidiagonal C* -Algebras
From the Bott periodicity theorem it follows that the K-theory of an
arbitrary A is isomorphic to that of a QD algebra (i.e., 82 A). Of course,
one loses the order structure (in the Ko case) and we aren't suggesting that
this is an enlightening reduction; it's just another cute observation.
In Exercise 7.1.4 we pointed out that every separable C-algebra is a
quotient of an RFD algebra. We can now improve that result in the cases
of nuclear and exact C -algebras.
Corollary 7.3.8. Let A be a separable nuclear {resp. exact) C-algebra.
There is a nuclear (resp. exact) RFD C -algebra B such that A is a quotient
of B.
Proof. Every algebra is a quotient of its cone, so we may assume that A is
QD. Exercise 7.1.3 then provides us with an RFD algebra B with ideal J
such that
0 ---+ J ---+ B ---+ A ---+ 0is short exact and has a c.c.p. splitting. We can now appeal to Exercises
3.8.1 and 3.9.8 to conclude that B is nuclear (resp. exact) whenever A is
nuclear (resp. exact). D
Exercise
Exercise 7.3.1. Given A, show that every faithful representation of either
CA or SA is quasidiagonal. (Hint: CA has no nonzero projections.)7.4. Two more examples
We've already seen that quasidiagonality of a reduced group C* -algebra im-
plies amenability. But what about the universal algebras? As you might
expect, things are very different (at least in the free group case).
Since C*(I') is defined by a representation-theoretic universal property,
one might expect that it is residually finite-dimensional (Definition 7.1.6)
whenever r is residually finite. This turns out to be false, in general, but
true for free groups.^9 However, it is not residual finiteness that explains this
phenomenon; it is universality of free groups.
Theorem 7.4.1 (Choi). The full group C* -algebra C* (lFn) is RFD for every
n = oo, 1, 2, 3, ....(^9) Bekka has shown that many residually finite groups do not have residually finite-dimensional
universal algebras ([14]). The proof is not so simple, though, and it would be nice to have an
elementary argument.