372 12. Approximation Properties for Groupsand Kirchberg [101] independently proved that an exact (or locally reflexive)
C* -algebra with the OAP has the SOAP.
Corollary 12.4.6. The class of algebras with the OAP is closed under ex-
tensions.Proof. Let 0 __, J -t B ~A -t 0 be a short exact sequence of C*-algebras
with J and A having the OAP. By Theorem 12.4.4, to prove that B has
the OAP, it suffices to check the slice map property for (B, II((.£^2 ), X), where
X c OC(.£^2 ) is an arbitrary closed subspace.
Let x E F(B, OC(.£^2 ), X) be given. Since A has the OAP, it follows that
(7r ® id)(x) E F(A, OC(.£^2 ), X) = A® X. Since X is ®-exact (as defined
in Section 3.9), there is a canonical isometric isomorphism A® X = (B ®
X)/(J®X) and (7r®id)(x) lifts toy E B®X. It follows that x-y E ker(7r®
idnqc2)) = J ® OC(.£^2 ) and x -y E F(J,OC(.£^2 ),X) = J ® X. Consequently,
xEB®X. D
Remark 12.4.7. Consider a nonsemisplit extension B of Cone(C~(IF'2)) by
OC(.£2)
0 ~ OC(.£^2 ) ---7--B ~ Cone(C~(IF'2)) ~ 0
(which will be shown to exist in Section 13.4). The C*-algebra B is not
exact, but it has the OAP by Corollary 12.4.6. It follows that the OAP does
not imply the SOAP.
Definition 12.4.8. We say a group r has the AP (approximation property)
if there exists a net^14 ( lfJi) of finitely supported functions on r such that
lfJi , 1 weak* in B2(r) - i.e., w(cpi) , w(l) for every w E Q(r). (See
Appendix D.)
It should be clear that the AP passes to subgroups and increasing unions.
Theorem 12.4.9. For a group r, the following are equivalent:
(1) the group r has the AP;
(2) the reduced group C* -algebra C~ (r) has the OAP;
(3) the reduced group C*-algebra C~(r) has the SOAP;
( 4) the group von Neumann algebra L(I') has the WOAP.Proof. We only prove the equivalence of (1) through (3). The implica-
tion (3) (2) is trivial and the proof of (2) (1) is similar to that of
Theorem 12.3.10, thanks to Lemma D.9. Thus, we must prove (1) * (3).
(^14) The Principle of Uniform Boundedness again implies that this net cannot be a sequence
unless I' is weakly amenable.