1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

380 13. WEP and LLP

(2) A@maxlffi(.€^2 ) = A@lIB(.€^2 ) canonically if and only if A has the LLP;

(3) C*(lF 00 ) @max B = C*(JF 00 )@ B if and only if B has the WEP.

Proof. Assertion (3) follows from Theorem 13.2.1 and Proposition 3.6.6.

The "only if" part of (2) follows from Corollary 13.1.4 and Theorem 13.1.6.

We will prove the "if" part of (2) and leave the proof of (1) to the reader.

Let A be a unital C* -algebra with the LLP. Let lF be a free group of

suitable cardinality and let 7r: C*(JF) -t A be a surjective *-homomorphism.

By the LLP, for any finite-dimensional operator system E, there is a u.c.p.

map 'ljJ: E -t C* (JF) such that 7ro'l/J = idE. It follows that the formal identity

E @ JIB ( .€^2 ) ~ C (JF) @ lffi ( .€^2 ) = C (JF) @max lffi ( .€^2 ) ~ A @max lffi ( .€^2 )
is contractive. Since E was arbitrary, we are done. D
Exercises
Exercise 13.2.1. Prove the following: If A is nuclear and B has the WEP,
then A @ B has the WEP; if A has the LLP and B is nuclear, then A @ B
has the LLP; if A is separable with the LP and Bis separable and nuclear,
then A @ B has the LP.

Exercise 13.2.2. Let A be a C -algebra with the WEP and let B =
C(JF 00 )/J. Prove that A @max B = (A@ C(JF 00 ))/(A@ J). Use tensor
products to give another proof of the fact that a C -algebra which is exact
and has the WEP is nuclear (Exercise 2.3.14).

13.3. The QWEP conjecture

We say a C -algebra A is Q WEP if it is a quotient of a C -algebra with
the WEP. Kirchberg's QWEP conjecture asserts that every C* -algebra is
QWEP, and it turns out to be equivalent to several seemingly unrelated open
problems, including Connes's embedding problem (statement (3) below).
Consequently, the QWEP conjecture is one of the most important open
problems in the theory of operator algebras.

Theorem 13.3.1 (Kirchberg). The following conjectures are equivalent:

(1) every C*-algebra is QWEP;

(2) C*(lFoo) @max C*(lF 00 ) = C*(lF 00 ) ® C*(lF 00 ) canonically;

(3) every type II1 -factor with separable predual is embeddable into the

ultraproduct Rw of the hyperfinite type II1 -factor R;

(4) the (separable) predual of any van Neumann algebra is isometrically

isomorphic to a subspace of the Banach space ultraproduct (Si)w of

the predual Si oflIB(.€^2 ).