14.2. Characterization of weak exactness 397Proof. Let M be a von Neumann algebra with the W*OAP and let l.fJi be
a net of finite-rank maps on M which converges to idM in the stable point-
ultraweak topology. Let J <l B, 7f and if be given as in Definition 14.1.1.
Recall that Q: B _,. B / J is the canonical quotient map. It follows from
Corollary 14.1.9 that the net 7f o (I.pi® idB) (x) converges ultraweakly to 7r(x)
for every x EM® B. Let x E ker(idM ® Q) be given. Since
(idM ® Q)((l.fJi ® idB)(x)) = (l.fJi ® idB/J)(idM ® Q)(x) = 0
and since I.pi is of finite-rank, we have (1.pi ® idB)(x) E M 0 J for all i. It
follows that
7r(x) = li:~n7f o (l.fJi ® idB)(x) = 0.
i
Therefore, ker(idM ® Q) C ker7r and if is min-continuous. 0
Exercises
Exercise 14.1.1. Let A be a C*-algebra such that A** is weakly exact.
Prove that A is exact if it is locally reflexive.
Exercise 14.1.2. Prove Corollary 14.1.5.
Exercise 14.1.3. Prove that a von Neumann algebra M is weakly exact if
and only if for any J <l B and any weakly nuclear u.c.p. map 1.p: B ,. M
with J C ker 1.p, the induced u.c.p. map rp: B / J ,. M is weakly nuclear.
Exercise 14.1.4. Let MC NC lffi(H) be von Neumann algebras with M
weakly exact. Suppose that there exists a u.c.p. map q): lffi(H) _,. N such
that q)IM = idM. Prove that the inclusion M '----' N is weakly nuclear. (It is
not known whether the weak exactness assumption on Mis necessary.)
14.2. Characterization of weak exactness
In this section, we will give an approximation characterization of weakly ex-
act von Neumann algebras, which is analogous to that of exact C -algebras.
First, we need a von Neumann algebra analogue of Proposition 9.2.7. Recall
that for von Neumann algebras M and N, a -representation 7f of M 0 N
(or M ® N) is said to be bi-normal if both 7f IM and 7f IN are normal.
Proposition 14.2.1. A van Neumann algebra Mis weakly exact if and only
if for any C* -algebra B and any left normal representation 7f: M ® B _,.
lffi(H), the bi-normal extension 11-: M 0 B** -7 IB(H) is min-continuous.
Proof. We first prove the "if" part. Let J <l B and 7f: M ® B ,. lffi(H) be
as in Definition 14.1.1. Let p be the central projection which supports the
normal *-homomorphism Q: B ,. (B / J), and identify (B / J) with
pB**. We denote the canonical inclusion by
'ljJ: B/J _,. (B/J) = pB c B**.