402 14. Weakly Exact von Neumann Algebras
1 1 1
:::; l~li;r1(2n + 2n-l + ... + 2m)[[a[[
=0,where the limits are taken along appropriate converging subnets. It follows
that 'ljJ o is the identity map on LJ En. Now extend to the norm closure
of LJ En, by norm continuity, and then further extend [A to : A -----+ S
by ultra weak continuity. Then, 'ljJ o : A* -----+ M is a normal u.c. p. map such
that 'ljJ o
which is the extension of the identity representation A '-----t M. Hence, the
restriction <p of to the (nonunital) von Neumann subalgebra M C A**
satisfies 'ljJ o <p = idM. D
It would be interesting to know whether or not one can choose S in
condition (3) to be an exact C-algebra (and, in addition, 'ljJ to be a normal
-homomorphism).
Proof of Theorem 14.1.6. Let r be a discrete group whose group von
Neumann algebra L(r) is weakly exact. We will verify condition (2) of
Theorem 5 .1. 6.
Let a finite subset E c r and c: > 0 be given. By Theorem 14.2.4 and
Arveson's Extension Theorem, there exist u.c.p. maps <p: L(r) -----+ Mn(C)
and 7/J: Mn(C)-----+ IB(£^2 (r)) such that for(}= 'ljJ o <p, one has
(}(>,(s)) E L(I') and [[.>-(s) - (}(>,(s))[[2 < c:
for every s EE. We may assume that cp(A.(s)) = 0 for all but finitely many
s E r. It follows that the kernel
u(s, t) = (e(A.(sr^1 ))A.tOe, AsOe)
is positive definite and satisfies [u(s, t) - 1[ < c: whenever sC^1 E E. This
completes the proof. D
Corollary 14.2.5. Let Ml and M2 be von Neumann algebras with separable
preduals. If both Mi are weakly exact, then Ml ® M2 is weakly exact.
Proof. By Theorem 14.2.4, there exist exact operator spaces Si and normal
u.c.p. maps 'Pi: Mi -----+ Si and 7/Ji: Si -----+ Mi such that 7/Ji o 'Pi = idMi.
Since Si is exact and has property C, the canonical bi-normal inclusion
1,: Si 0 S-2 '-----t (S1 ® S2) is a continuous u.c.p. map on the minimal tensor
product. We first consider 'ljJ = (7/J1 ® 7/J2)[s 1181 s 2 and extend it to a normal
u.c.p. map, still denoted by 1/J, from (S1 ® 82) into Ml® M2. We have
'ljJ o /, = 7/J1®1/J2 since both maps are bi-normal and agree on S 1 ® S 2 • Let
(/;: (M1 ® M2) -----+ (S1 ® S2) be the normal extension of the u.c.p. map
1,o(cp1®<p2): Ml®M2-----+ (S1®S2). Then, 1/Jo(j;: (M1®M2)-----+ Ml®M2