1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

396 14. Weakly Exact von Neumann Algebras

Remark 14.JL.7. Since there exists a discrete group which is not exact [71]

(see also Remark 5.5.10), there exists a von Neumann algebra which is not

weakly exact. However, it would be very nice to have simpler examples; a

prime candidate is Rw.

A large number of von Neumann algebras have the W*OAP (see Sec-

tion 12.4). We now show that the W*OAP implies weak exactness. Denote

by CB(M) the Banach space of all completely bounded maps on M and

define

CB(M) =the closed linear span of {wa,f: a EM, f EM} C CB(M)*,

where Wa,J(lf>) = f(tp(a)) for <p E CB(M). For a E M 0 IIB(£^2 ) and f E

(M 0 IIB(£^2 ))*, we define Wa,f E CB(M)* with llwa,111 S': llall llfll by^2

Wa,J(lf>) = f(<p ® idJIB(£2)(a)).

The proof of the following proposition is very similar to that of Lemmas D.7-

D.9, and hence it is omitted.

Proposition 14.1.8. For a von Neumann algebra M, the following are

true.

(1) One has Wa,f E CB(M)* for every a E M 0 IIB(£^2 ) and f E (M 0

IIB(£^2 ) k

(2) There is a canonical isometric isomorphism CB(M) = (CB(M)*)*.

(3) Every element in CB(M)* is of the form Wa,f with a E M ® JK(£^2 )

and f E (M 0 IIB(£^2 ))*.

In particular, the stable point-ultraweak topology (Definition 12.4.1) coin-

cides with the O"(CB(M), CB(M)*)-topology.

Corollary 14.1.9. Let B be a C* -algebra, x E M ® B and f E (M ® B)*.

If f( · ® b) E M* for every b E B, then Wx,f E CB(M)*, where Wx,J(lf>) =

f((<p ® idB)(x)) for all <p E CB(M).

Proof. Let (xn) be a sequence in M 8 B which converges to x. It is clear
that the sequence (wxn,J) is in CB(M)* and converges to Wx,f· D

Proposition 14.1.10. A von Neumann algebra M with the WOAP is
weakly exact.

(^2) The map cp 181 idJIB(£2) is defined as the point-ultraweak limit of cp Q9 Wn, where Wn is the
compression onto JIB(£;). It is uniquely determined by the identity (idM Q9 g) o (cp Q9 id1E(£2)) =
cp o (idM Q9 g) for all g E JIB(£^2 )*.