1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.4. Another approximation property 371

Proof. We only prove assertion (2). Denote by F(A) the set of all con-

tinuous finite-rank linear maps on A. Note that every <p E F(A) is of the

form cp( ·) = :E~=l wk(· )ak for some n E N, wk E A* and ak E A. For

x EA® B, we set Rx= {(w ® idB)(x) : w EA*} CB (which may not be

closed). It is easy to see that {(c.p®idB)(x): <p E F(A)} =AG Rx and that

x E F (A, B, X) if and only if Rx c X.

Now suppose that A has the SOAP and x E F(A, B, X) is given. We

may assume that B is separable since for every x E A® B one can find a

separable C*-subalgebra Bo c B such that x E A® Bo. Let ('Pi) be a net

of finite-rank continuous linear maps on A such that the net ('Pi® idlffi(£2))

converges pointwise to the identity map on A®IBl(.€^2 ). Since B <=--+ IBl(.€^2 ), we

have x = lim(<pi ® idB)(x) EA® X.

To prove the converse, let x1, ... , Xn E A® IBl(.€^2 ) and E > 0 be given.

Now set

n n

x = diag(x1, ... , Xn) E EEJ(A ® IBl(.€^2 )) CA® IBl(E!jR^2 )

k=l k=l

and let X =norm-cl Rx be the norm closure of Rx. If the triple

satisfies the slice map property, then

x E F(A,IBl(E!jR^2 ),X) =A® X = norm-cl{(c.p ® idlffi(E9£2))(x): <p E F(A)}

and hence there exists <p E F(A) such that llxk - (cp ® idlffi(E9£2))(xk)ll < E
for every k. D

Remark 12.4.5. In the proof above, we have not used the C* -structure
of A at all - i.e., the theorem, except the statement that SOAP implies
exactness, holds for an arbitrary operator space.

We should also mention that Tomita's celebrated commutation theorem
is equivalent to the validity of the weak slice map property for arbitrary
von Neumann algebras (M, N, B). The standard formulation of Tomita's
result ([183, Theorem IV.5.9]) is that (M ® N)' = M' ® N' in IBl(H ® K),
for arbitrary van Neumann algebras M C IBl(H) and N C IBl(K:). But it is
easy to see that

Fu(M, N, B) c (M ® IBl(K)) n (IBl(H) ® B)

= (M' ®Cl)' n (Cl® B')' = (M' ® B')'.

Finally, it is a long-standing open problem whether or not there ex-

ists an exact C* -algebra without the OAP. The reduced group C* -algebra

C~ (SL(3, Z)) is a candidate [80]. We note that Haagerup and Kraus [80]