14.2. Characterization of weak exactness 401(3) =?-(1): The proof is virtually identical to that of Theorem 14.1.2.
(2) =?-(3): Let M be a weakly exact von Neumann algebra with separa-
ble predual and A be an ultraweakly-dense norm-separable C* -subalgebra in
M. We denote by II · 110" a norm which defines the ultraweak topology on the
unit ball of M. Using condition (2) recursively, we can construct sequences
of finite-dimensional operator systems and connecting u.c.p. maps to get a
diagramsuch that
(1) El C E2 C ···CM and the norm closure of U En contains A,
(2) Fn C Mk(n)(C) for some k(n) EN,
(3) the diagram commutes and llBn(a) - all<T < 2-nllall for a E En.We define the operator system S to be the inductive limit of (Fn, O"n); more
precisely, let Q: IJMk(n)(CC) -t IJMk(n)(C)/ ffiMk(n)(C) be the quotient
map and let S = Q(S), where Sis the norm closure of
00
{(xn)n E IJ Mk(n)(CC): ::lm with Xm E Fm and Xn+l = O"n(xn) for n 2: m}
n=l
~ LJ (Mk(1)(CC) EB··· EB Mk(m-1)((['.) EB Fm)·
m2:1
Since the operator system Sis exact (and locally reflexive), so is the quotient
Q(S) by the "ideai" ffiMk(n)(C). (That's an exercise.) Let : LJEn -t S*
be a point-weak cluster point of the sequence
<I>m: Em 3 X 1---t Q((O"n-1 o · · · o O"m o <?m(x))~=m) ES CS**
and let;/;: S -t M be a point-weak* cluster point of the sequence
;/Jm: S 3 (xn)n 1---t 'fm(xm) EM.It is not hard to see that ;/; = 'ljJ o Q for some u.c.p. map 'ljJ: S -t M. The
unique weak* -continuous extension of 'ljJ on S** is still denoted by 'ljJ. Then,
for any a E LJ En, we have that
Ila - ('f o <I>)(a)ll<T =Ila - lim m ('f o <I>m)(a)il<T
=Ila - limlim m n ('fn o O"n-1 o · · · o O"m o <?m)(a)ll<T
= Ila - limlim m n (Bn o · · · o Bm)(a)ll<T