1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

384 13. WEP and LLP

Of course, if C~(r) is QWEP, then so is L(r), by Lemma 13.3.6. But

typically it's very hard to show reduced group C* -algebras are QWEP. Here's

one case where it can be done.

Proposition 13.3.8. The reduced group C* -algebra C~ (IFr) of the free group

IF r is relatively weakly injective in the group van Neumann algebra L(IF r).

In particular, C~ (JF r) is Q WEP.

Proof. By Lemma 13.3.4, it suffices to show that C~ (IF r) is relatively weakly

injective in L(IFr)· For this, we will verify condition (2) in Lemma 13.3.2.

Recall that IFr is weakly amenable with Acb(IFr) = 1 (Corollary 12.3.5).

Hence, there exists a sequence of finitely supported functions 'Pn on IF r which

converge pointwise to 1 and such that their multipliers m'Pn are (completely)

contractive on L(IFr)· Since each m'Pn maps L(IFr) into C~(IFr) and since the

sequence { m'Pn} converges to the identity on C~ (IF r), we are done. 0

A variant of the proof above shows that the reduced group C* -algebra
C~ (r) is QWEP provided that r is residually finite and weakly amenable.
However, it is not known whether or not C~(SL(3, Z)) is QWEP. Actually,
the following question appears to be open: Does there exist a group r such
that C~(r) is not relatively weakly injective in L(I')?

Exercises
Exercise 13.3.1. Let A and B be unital C-algebras and r be a tracial state
on A 8 B such that 7r 7 (A 8 B)" is a factor. Prove that r = (r[A) ® (r[B)·
Exercise 13.3.2. Let C be a unital C -algebra and T be the set of tracial
states on C. Prove that r ET is an extreme point if and only if 7rr(C)" is
a factor.
Exercise 13 .. 3.3. Let A and B be unital C -algebras and r be a tracial
state on A 8 B. Prove that r is continuous on A® B. (This is equivalent to
asserting that any -homomorphism from A 8 B into a finite von Neumann
algebra is continuous on A ® B.)

Exercise 13.3.4. Prove that C(IF 00 ) ®max C(IF 00 ) = C(IF 00 ) ® C(IF 00 )
canonically if and only if C* (IF 00 x IF 00 ) has a faithful trace.

Exercise 13.3.5. Let Ci ( i = 1, 2) be C* -algebras with the LLP and Ji be
a closed two-sided ideal in Ci. Prove that if Ai= Ci/ Ji is QWEP, then

Ai ®max A2 =Ci® C2/(Ji ® C2 +Ci® h)

canonically.

Exercise 13.3.6. Prove that a finite von Neumann algebra M with separa-
ble predual is embeddable into Rw if it is QWEP. (Hint: Combine Exercise
13.3.5 and Theorem 6.2.7).