1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

370 12. Approximation Properties for Groups

Remark 12.4.2. A Banach space Xis said to have the AP (approximation
property) if there exists a net of finite-rank continuous linear maps 'Pi on X
such that 'Pi ---+ idx uniformly on compact subsets. We note that 'Pi ---+ idx
uniformly on compact subsets if and only if 'Pi ® idco(N) converges to the
identity map on X@ co ~ co(N; X) in the point-norm topology. Since
c 0 (N) c JK(f^2 ), the OAP implies the AP. Szankowski [180] proved that
lll(£^2 ) does not have the AP. It follows that lll(£^2 ) does not have the OAP.

Though we've already encountered them, it seems fitting to properly
introduce slice maps on tensor products: Given C -algebras A and B and a
linear functional w E A, the map w ® idB: A® B ---+ C ® B ~ B is called a
slice map.

Definition 12.4.3. Let A and B be C* -algebras and X C B be a closed

subspace. We define a subset F(A, B, X) c A® B (Fis for Fubini) by

F(A,B,X) = {x EA® B: Vw EA*, (w ® idB)(x) EX}.

We say a triple (A, B, X) satisfies the slice map property if F(A, B, X) =
A® X, the norm closure of A 0 X in A® B.
Let Mand N be von Neumann algebras and X c N be an ultraweakly
closed subspace. Define a subset F(Y(M, N, X) c M ® N by

F(Y(M,N,X) = {x EM® N: Vw EM*, (w ® idN)(x) EX}

and we'll say the triple (M, N, X) satisfies the weak slice map property if

F(Y(M, N, X) is the ultraweak closure of M 0 X in M ® N.

One should check that for an ideal J of B, the triple (A, B, J) satisfies
the slice map property if and only if the sequence

0 ____,._ A ® J ____,._ A ® B ____,._ A ® (BI J) ____,._ 0

is exact.

Theorem 1~~.4.4. The following are true:

(1) a C*-algebra A has the OAP if and only if (A, lli:.(£^2 ), X) satisfies

the slice map property for any closed subspace X c lli:.(£^2 );

(2) a C*-algebra A has the SOAP if and only if (A, B, X) satisfies the

slice map property for any B and any closed subspace X c B;

(3) a von Neumann algebra M has the VV*OAP if and only if (M, N, X)

satisfies the weak slice map property for any N and any ultraweakly

closed subspace X c N.

In particular, the SOAP implies exactness.