394 14. Weakly Exact von Neumann Algebraswith M@ J C ker7r, the induced representation ii': M 0 (B / J) ----+ llll(H) is
min-continuous.Note that by Theorem 3.8.5, a left normal -representation 7f: M 0 B ----+
llll(H) is min-continuous if and only if the -homomorphism 7r[B: B----+ 7r(M@
CClB)' is weakly nuclear.
Theorem 14.1.2. A van Neumann algebra Mis weakly exact if it contains
a weakly dense C* -algebra A which is exact.
Proof. Let A be an exact C*-algebra such that M =A". By Theorem 9.3.1,
A is locally reflexive. Let J <I B, 7f and 7f be given as in Definition 14.l.l.
Since A is exact, 7f is continuous on A @ ( B / J). Since local reflexivity is
equivalent to property C" (Proposition 9.2.5), ii' extends to a left normal
*-representation p on A** ® (B / J). Let p E A** be the central support
projection of P[A**: A** ----+ M (so that we may identify M with pA**). We
denote this identification by i: M ----+ pA ** C A**. It follows that 7f coincides
with po (i@ id), which is min-continuous. D
Remark 14.1.3. Suppose that the second dual A** of a C*-algebra A is
weakly exact. Then, A is exact if and only if it is locally reflexive (Exercise
14.1.1). The assumption of local reflexivity is essential. Indeed, there exists
a nonexact C* -algebra whose second dual is weakly exact. For instance, an
extension of exact C*-algebras need not be exact (see Theorem 13.4.1), but
it is easily checked that the direct sum of two weakly exact van Neumann
algebras is again weakly exact. It is not known whether there exists a locally
reflexive C* -algebra which is not exact.
Proposition 14.1.4. Let M be a van Neumann algebra. Suppose that there
exists a net lvfi of weakly exact van Neumann algebras and nets of normal
c.p. contractions 'Pi: M ----+ Mi and 'I/Ji: Mi ----+ M such that the net 'I/Ji o 'Pi
converges to the identity on M in the point-ultraweak topology. Then, M is
weakly exact.Proof. Let J <I B and a left normal -representation 7f: M@ B ----+ llll(H)
with M ®JC ker7r be given. We consider the left normal c.p. contraction
Wi = 7r o ('I/Ji® idB): Mi® B----+ llll(H).
Since Mi 3 a~ Wi(x(a ® lB)Y) E llll(H) is ultraweakly continuous for every
x, y E Mi @ B, the minimal Stinespring dilation of wi is a left normal -
representation which still vanishes on Mi®J. (One should verify this.) Thus,
it follows from weak exactness of Mi that the induced c.p. map '1ri: Mi 0
(B / J) -+ JE('H) is min-continuous. Since 7f is left normal, the net of c.p.
contractions
'1ri 0 ('Pi@ idB/ J): M@ (BI J) ----+ llll(H)