1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

14.2. Characterization of weak exactness 399

Corollary 14.2.2. Let M be a weakly exact von Neumann algebra and X

be an operator space. Then, for any z E M 0 X** with llzllmin :::; 1, there

exists a net (zi) in M 0 X with llzillmin :::; 1 which converges to z in the

O'(M Q9 X**, M* 0 X*)-topology.

Proof. Let z EM 0X be such that llzllmin:::; 1. By the above discussion,
we have ll8x(z)ll:::; 1. Hence, there exists a net (zi) inM0Xwith llzillmin:::;
1 which converges to 8x(z) in the weak*-topology. We claim that the net
(zi) converges to z in the O'(M Q9 X*, M 0 X)-topology. Let f E M and
g E X be given. We'll write J for f when it is regarded as an element
in M. Then, f(a) = f(7r(a)) for a E M, where 7r: M -+ M is the
normal extension of the identity on M. (We have identified M = pM and
7r(a) =pa.) Now, J Q9 g E (M Q9 X)* and

(! Q9 g)(8x(a Q9 x)) = f(pa)g(x) = f(a)g(x) = (! Q9 g)(a Q9 x)

for every a E M and x E X**. Therefore, we have

(! Q9 g)(z) = (! Q9 g)(8x(z)) = li~(f Q9 g)(zi) = li~(J Q9 g)(zi)

i i

as claimed. D

Operator space duality (Theorem B.13) immediately yields the following
reformulation.

Corollary 14.2.3. Let M be a weakly exact von Neumann algebra and X

be an operator space. Then, for any finite-rank c. c. map <p : X* -+ M, there

exists a net ('Pi) of weak* -continuous finite-rank c. c. maps 'Pi: X* -+ M

which converges to <p in the point-ultraweak topology.

We need the notion of exactness for operator systems. An operator
system S is said to be exact (or more precisely 1-exact) if ( S Q9 B) / ( S Q9 J) =
S Q9 ( B / J) canonically isometrically for any J <I B. All the characterizations
of exactness for C* -algebras (Section 3.9) are still valid for operator systems,
with obvious modifications (one can verify this or see [61]). Moreover, exact
operator systems are locally reflexive in the sense of Definition 9.1.2 (cf.
Theorem 9.3.1) - although this is not so easy to prove (see [61] for the
details).

The following is a von Neumann algebra analogue of Theorem 3.9.1 and
a partial converse to Theorem 14.1.2. The implication (1) {:? (2) holds
without the separability assumption, by replacing sequences with nets.

Theorem 14.2.4. Let M be a van Neumann algebra with separable predual.
The fallowing are equivalent:
(1) M is weakly exact;