400 14. Weakly Exact van Neumann Algebras(2) for any finite-dimensional operator system E in M, there exist se-
quences of u.c.p. maps Cfi: E ----+ Mn(i) (<C) and 'I/Ji: Cfi(E) ----+ M
such that the net ('I/Ji o Cfi) converges to idE in the point-ultraweak
topology;
(3) there exist an exact operator system S and normal u.c.p. maps
cp: M ----+ S**, 'lj;: S** -> M such that 'lj; o cp = idM.Proof. (1) =? (2): Let M be a weakly exact von Neumann algebra acting on
a Hilbert space H and let E c M be a finite-dimensional operator system.
For simplicity we assume'}{= £^2 and we let n: E----+ Mn(<C) be compression
by the projection onto £~ C £^2. Let En = n(E) and = EB n: E ----+
IT En. We note that is a complete isometry and IT En C IT Mn(<C) is an
ultraweakly closed operator subspace. A left inverse of is given by
'1!: II En 3 (xn)~=l f---+ n-tw lim ;;:-^1 (xn) EEC M,
where w is a fixed free ultrafilter on N. The limit exists since E is finite-
dimensional and n is injective for large n. Moreover, we have
llidMk(C) 0 '1!11 ::::; n-tw lim ll(idMk(C) 0 n)-^1 11=1
for every k. Hence ]! is a complete contraction with '1! o = idE. By
Corollary 14.2.3, there exists a net of ultraweakly continuous c.c. maps
'1! i : IT En ----+ M such that limi '1! i = '1! in the point-ultra weak topology.
Since each ]! i is ultraweakly continuous, if we set
N N oo
CfN =EB n: E----+ EB En C II En
n=l n=l n=l
and 'l/Ji,N = '1!il(fN(E), then we have
lim lim 'l/Ji,N 0 Cf N = li:μi ]!i 0 = ]! 0 = idE.
i N-too i
Since CfN(E) C EB;;=l En C EB;;=l Mn(<C), this implies condition (2) except
that the 'l/Ji,N's may not be u.c.p. One can fix this problem as follows. First,
passing to a convex combination, one may assume convergence in the point-
ultrastrong topology. Then, arguing as in the proof of Proposition 3.8.2, one
may further assume that 'l/Ji,N is approximately unital in norm. Finally, one
invokes Corollary B.11 to perturb 'l/Ji,N to u.c.p. maps.
( 2) =? ( 1): Let J <JB, 7r and ii" be given as in Definition 14.1.1. Let E c M
be a finite-dimensional operator system and take nets (cpi) and ('I/Ji) as in
condition (2). Since Ei = Cfi(E) C Mn(i)(<C), we have (Ei 0 B)/(Ei 0 J) =
Ei 0 ( B / J) isometrically. It follows that ii" o ('I/Ji 0 id B; J) is contractive
on Ei 0 (B / J). Since ii" is left normal, the net ii" o ( ('I/Ji o Cfi) 0 id B /J) of
contractions converges to ii" on E 0 (B / J). Since E c M is arbitrary, ii" is
contractive on M 0 ( B / J).