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(4) the induced map a*: Ko(A)------+ Ko(A) "compresses no element." In
other words, if x E Ko(A) and a*(x) :::::; x (in the natural order on
Ko(A)), then a*(x) = x.Katsura has studied the AF embeddability of Cuntz algebras by certain
actions of JR in [98].
Regarding the AF embeddability of general exact RFD algebras, we have
the following striking result of Dadarlat ([52]).
Theorem 8.5.5. Assume A is separable exact RFD and satisfies the uni-
versal coefficient theorem (UCT) of Rosenberg and Schochet ([170]). Then
A is AF embeddable.In particular, every type I C -algebra with a faithful tracial state is AF
embeddable (since these are RFD by Proposition 7.1.8). A long-standing
open question asks whether or not every nuclear C -algebra satisfies the
UCT. An affirmative answer would imply every nuclear RFD algebra is AF
embeddable (or, if you prefer, the construction of an RFD nuclear algebra
which is not AF embeddable would provide a counterexample to the UCT).
The last result we'll mention requires a definition.Definition 8.5.6. A C*-algebra is called residually finite if every quotient
is stably finite.
This notion is completely unrelated to residual finite-dimensionality (i.e.,
neither implies the other) and is much stronger than being stably finite. For
example, the cone over any algebra is stably finite but often fails to be
residually finite.
In [179] Spielberg proved the following:Theorem 8.5. 7. Every residually finite type I C* -algebra is AF embeddable.
A natural question, pointed out to us by Kerr, is whether the previous
result can be extended to stably finite type I C* -algebras. Here are some
related open problems.
Open Problems
(1) (Blackadar-Kirchberg) Is a stably finite nuclear C*-algebra neces-
sarily QD?
(2) (Blackadar-Kirchberg) Is every nuclear QD C*-algebra AF embed-
dable?
(3) Does every exact QD C*-algebra embed into a nuclear QD C*-
algebra?