13.3. Tb.e QWEP conjecture 383Since A has the LP, there is a c.c.p. lifting 'ljJ: A--"* B. Let A** c IIB('H) be
a universal representation. Since B has the WEP, the c.c.p. map 'ljJ extends
to a c.c.p. map if;: IIB(H) --"* B**: ·A~~-~A**
/~
/ /
/
'!/; JIB ( H) 7r**~
B~-----~ B.
It follows that cp = 7r o if; is a c.c.p. map such that cplA = 7f o 'ljJ = idA.
(2) =? (3): Let M be a finite van Neumann algebra with separable
predual and T be a faithful normal trace on M. Let 7f: C(IF 00 )--" M be a-
homomorphism with ultraweakly dense range. By assumption and Theorem
6.2.7, the trace To 7f is an amenable trace on C(IF 00 ). It follows that there
is a -homomorphism p: C(IF 00 )--" Rw such that Two p =To 7f. It is easy
to see that the van Neumann subalgebra in Rw generated by p( C (IF 00 )) is
-isomorphic to M (cf. Exercise 6.2.4).
(3) =? (1): The C-algebra Rw is QWEP by construction. Let M be a
van Neumann subalgebra in Rw. Since every van Neumann subalgebra in a
finite van Neumann algebra is the range of a conditional expectation, M is
also QWEP by Lemma 13.3.4. Since every finite van Neumann algebra with
separable predual is embeddable into a type II1-factor with separable predual
(e.g., take a tracial free product with the hyperfinite !Ii-factor R, [154]),
every finite van Neumann algebra with separable predual is embeddable
into Rw by assumption and is QWEP. It follows from Lemma 13.3.6 that all
semifinite van Neumann algebras are QWEP. By Takesaki's duality theorem
(Theorems 9.3.5 and 9.3.7) we conclude that all van Neumann algebras are
QWEP. In particular, all double duals are QWEP, so all C*-algebras are
QWEP, too. D
Modifying the proof of (1) =? (2), it can be shown that a QWEP C*-
algebra with the LLP also has the WEP. Hence, the QWEP conjecture is
equivalent to saying that the LLP implies the WEP. There is no known
example of a nonnuclear C* -algebra with both the WEP and the LLP.
From the proof of (3) =? (1), we see that a finite van Neumann algebra
with separable predual is QWEP if it is embeddable into Rw. (The converse
is also true.) Hence we obtain the following corollary, the proof of which is
left to the reader.
Corollary 13.3. 7. Let r be a group with Kirchberg's factorization property.
Then, the group von Neumann algebra L(I') is QWEP.