280 8. AF Embeddability
Since 000 contains a copy of Co(O, 1], so does B. Thus, if A is exact, we
can invoke Kirchberg's embedding theorem (Theorem 10.2.2) to deduce
Co(0,1] @Ac Co(O, 1] @0 00 c B@0 00 ~ B.
This yields Theorem 8.3.5, since B is AF embeddable.
As previously noted, the Pimsner-Voiculescu AF embedding of irrational
rotation algebras first sparked interest in the subject. Hence it was natural
to consider AF embeddability of more general crossed products and this has
been studied by many authors. The results tend to be quite difficult and we
really don't have a very good understanding of general crossed products at
this point. However, for actions of Z there are two cases where the picture
is clear. In [145] Pimsner proved the following theorem.
Theorem 8.5.3. Let X be a compact metric space and a E Aut(C(X)) be
an automorphism. Then the following are equivalent:
(1) C(X) ><la Z is AF embeddable;
(2) C(X) ><la Z is QD;
(3) C(X) ><la Z is stably finite;
(4) if ha : X -+ X is the homeomorphism corresponding to a, then
"ha compresses no open sets." That is, if U c X is open and
ha(U) CU, then U = ha(U), where U is the clqsure of U.^4
Related results for actions of other groups on abelian C* -algebras have
been found by Pimsner (actions of JR - see [14 7]) and Matui (actions of Z^2 -
see [124]). See also [118] and [119], and the references therein, for related
results of Lin.
Voiculescu first studied the AF embeddability question for crossed prod-
ucts of AF algebras by the integers in [188]. His results, together with im-
portant Rohlin property contributions of Kishimoto and others, played an
important role in the following ([26]):
Theorem 8.5.4. Let A be an AF algebra and a E Aut(A) be an automor-
phism. Then the following are equivalent:
(1) A ><la Z is AF embeddable;
(2) A ><laZ is QD;
(3) A ><la Z is stably finite;(^4) A more dynamically pleasing formulation of this condition can be given in terms of "pseu-
doperiodic" points - namely, every point should be pseudoperiodic. By definition a point xo E X
is pseudoperiodic if for every e > 0 there exist points x1, ... , Xn such that d(xi+i, ha(xi)) < e
(where dis a fixed metric on X) for i E {O,.. ., n - 1} and d(xo, ha(xn)) < e as well.