13.4. Nonsemisplit extensions 385Exercise 13.3.7. Use the previous result and Exercise 6.2.4 to show that
the QWEP conjecture is equivalent to knowing that every trace on C*(JF 00 )
is amenable.
Exercise 13.3.8. Prove that if r is a countable group with Kirchberg's
factorization property, then L(r) is embeddable into Rw.
Exercise 13.3.9. Prove that a C*-algebra which is locally reflexive and
which has the WEP is nuclear.13.4. Nonsemisplit extensions
This section is devoted to the following theorem:
Theorem 13.4.1 (Kirchberg). Let A be a separable QWEP C-algebra and
CA = Co(O, 1] ®A be the cone over A. Then, there exists a quasidiagonal
extension (Definition 10. 3. 3)
0 ,,.. lK ( 1!^2 ) ,,.. B ,,.. CA ,,.. 0
such that B has the WEP.
Remark 13.4.2. Arveson's Extension Theorem implies that if an extension
0 -----+ J -----+ C -----+ D -----+ 0 is locally split and C has the WEP, then D also has
the WEP. Since there is always a splitting for the quotient map CA -----+ A,
it follows that if A in the theorem above does not have the WEP, then
the sequence 0 -----+ JK(R^2 ) -----+ B -----+ CA -----+ 0 can't be locally split. In the
case A = C~(lF2), which is QWEP (Proposition 13.3.8) but does not have
the WEP (see Exercise 13.2.2), we can apply the theorem and deduce that
the resulting algebra B is not locally reflexive, hence not exact. Thus, an
extension of exact C -algebras need not be exact and exactness cannot be
characterized in terms of double duals.^1
It is not known whether or not in the theorem one can take B to have
the LLP.
Turning to the proof of the theorem, let's ease notation by letting F
denote the full free group C -algebra C (JF 00 ). Let A be a separable unital
C -algebra and J be a closed two-sided ideal in C such that A = F / J. We
regard Fas the subalgebra of constant functions in C[O, 1] ® F (which is in
the multiplier algebra of the cone CF). Fix a quasicentral approximate unit
{en} of J in C (JF 00 ) and define *-homomorphisms Pn by
Pn: Co(O, 1] 3 fr-+ f(t ® (1-en)) E CF,
where t E Co(O, 1] is the identity function on (0, l].
lAnother example of a nonsemisplit extension of exact C*-algebras was given by Haagerup
and Thorbjprnsen who proved that C~ (IB'2) <-+ f1Mn(<C)/ E9Mn(IC).