1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

13.3. The QWEP conjecture 381

We will only prove the equivalence of (1), (2) and (3); see [134] for the

last condition (and the exercises at the end of this section for a few more

equivalent statements). We again need several lemmas. First, we add a local

characterization of "relative weak injectivity" to Proposition 3.6.6.

Lemma 13.3.2. Let A C B be C* -algebras. The following are equivalent.

(1) the C* -algebra A is relatively weakly injective in B - i.e., there

exists a c.c.p. map cp: B---+ A** such that 'PIA= idA;

(2) for any finite-dimensional subspace E C B and any c; > 0, there

exists a contraction 'ljJ: E---+ A such that 117/JIEnA - idEnAll < c.

Proof. The implication (1) =?-(2) follows from the principle of local reflex-

ivity for Banach spaces (Theorem 9.1.1). To prove the converse, assume

that for every finite-dimensional subspace E c B and c; > 0, there exists a

contraction 7/JE,c:: E---+ A such that 117/JIEnA - idEnAll < c;. Let cp: B---+ A**

be a cluster point of 7/JE,c: in the point-weak* topology. Then, cp is a contrac-

tion with 'PIA = idA. The weak* -continuous extension of cp on B** is still

denoted by cp. Then, cp: B** ---+A** is a contraction such that 'PIA** = idA**-

By Tomiyama's Theorem (Theorem 1.5.10), cp is a conditional expectation,

hence c.c.p. D

Lemma 13.3.3. Let {A} be a family of C*-algebras with the WEP. Then

their product IT Ai also has the WEP.

Proof. Let Ai c JB(Hi). Since IT JB(1ii) is injective, it is enough to verify
condition (2) in Lemma 13.3.2 for ITAi C ITlB(Hi)· Let E C ITlB(1ii)
be a finite-dimensional subspace and let c; > 0. Then, there exist finite-
dimensional subspaces Ei C JB(Hi) such that E C IT Ei. Since Ai has the
WEP for each i, there exists a contraction 'I/Ji: Ei---+ A such that 117/JilEinAi-
idEinAJI < c;. We define a contraction 'ljJ: E---+ IT Ai by 7/J((xi)i) = (7/Ji(xi))i
for (xi)i EEC ITEi· Evidently we have 117/JIEnITA - idEnITAJ < c. D
Lemma 13.3.4. Let Ao C A be C -algebras such that Ao is relatively weakly
injective in A. If 7r: B ---+ A is a surjective -homomorphism from a C* -
algebra B, then Bo= 7r-^1 (Ao) is relatively weakly injective in B. In partic-
ular, if A is Q WEP, then so is Ao.

Proof. Since Ao is relatively weakly injective in A, there exists a (normal)

conditional expectation 'ljJ from A** onto A(;*. If we set J = ker 7r, then

B 0 * ~ J** EB A~* c J** EB A**~ B**.

Hence, idJ** EB 'ljJ is a conditional expectation from B** onto B 0 *. If A is

QWEP and B has the WEP, then Bo has the WEP and Ao is QWEP. D