1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

9.2. Tensor product properties 291

<p: E --+A such that for every x E E and f E F,

lf(1.p(x) - x)I < c:llJll llxll-

We regard Mn(E) as a finite-dimensional subspace of Mn(A)** (via the

isometric identification Mn(A**) = Mn(A)**). It follows from the principle

of local reflexivity for Banach spaces (Theorem 9.1.1) that there exists a

net of contractions from Mn(E) into Mn(A) which converges to the identity

on Mn(E) in the point-weak* topology. Passing to convex combinations if

necessary, we may assume that this net converges in the point-norm topology

on Mn(ClA)· Hence, we may find a contraction Wn: Mn(E)--+ Mn(A) such

that ll'1rn1Mn(C1A) - idMn(IClA) II < 1/n and

1

l(w Q9 f)('1rn(x) - x)I < -llwllllfllllxll n

for every x E Mn(E), w E Mn(C)* and f E F. We set

~n(x) =Jr r u'lrn(uxv)v dudv,
Ju(n)xU(n)
where U(n) is the unitary group of Mn(C) (which is compact). It follows
that ~n is an Mn(C)-module map, or equivalently, there is t/Jn: E--+ A such
that idMn(IC) Q9 t/Jn = ~n· It is not hard to check llt/Jn(l) - lll < 1/n and

If ( t/Jn(x) - x) I < 2_ n llJ 11 llxll

for every x E E and f E F. Let Ao c A be a separable C* -algebra containing

all the subspaces t/Jn ( E), and let tjJ: E --+ A 0 * be a cluster point of the

sequence {t/Jn}. Then, t/J is unital and c.c. since llidMk(IC) Q9 t/Jnll :S ll~nll :S 1

for n 2:: k. Moreover, UIAo)(tjJ(x)) = J(x) for every x EE and f E F. Since

Ao is locally reflexive by assumption, the u.c.p. map tjJ: E --+ A 0 * can be

approximated by u.c.p. maps <p: E --+ Ao c A. D

We end this section with a technical result that will be needed in later

chapters.

Lemma 9.2.9. Let M and N be van Neumann algebras with weakly dense

C* -subalgebras B C M and C C N, respectively. Let <I>: M 8 N --+ lffi(H) be

a bi-normal^2 u.c.p. map. If B has property C and <I> is min-continuous on

B 8 C, then <I> is min-continuous on M 8 N.

Proof. We may assume that B and C are unital. Since is continuous
on B Q9 C, it extends to a normal u.c.p. map~ on (B Q9 C). Since B has
property C, the u.c.p. map~ is bi-normal and min-continuous on B 0C.
Let e E B and f E C be the central projections such that M = eB