1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.8. Nuclearity and tensor products gg

Exercise 3. 7.3. Let A be nonunital and assume that

0------* J@A-----*B@A-----*B/J@A-----*O

is exact. Show that the same sequence with A replaced by its unitization is

also exact.

3.8. Nuclearity and tensor products

In this section we come back to the historical origins of nuclearity, recasting
everything in terms of tensor products. The main theorems are not trivial
and require a few technical preliminaries. The first three requisite results
can be found in most basic operator algebra texts and may be known to
the reader. However, experience suggests that they are unfamiliar to many;
hence we'll state them as separate propositions. In order to keep sight of
the forest, we recommend skipping the proofs of these trees at first; after
the main results are absorbed, one can return and fill in the gaps.

Lemma 3.8.1. Let IIB(IIB('h'.)) be the set of bounded linear operators on IIB('h'.)

and C c IIB(IIB('h'.)) be any convex set. Then the point-weak operator topology

and point-strong operator topology closures of C coincide.

Proof. Since a convex subset of IIB('h'.) has the same closure in the weak or
strong operator topology, the proof of this lemma is very similar to Lemma
2.3.4. D
Proposition 3.8.2. Let A be a unital C -algebra, M be a van Neumann
algebra and cp: A------ M be a u. c.p. map. If cp belongs to the point-ultraweak
closure of the factorable maps (Definition 2. 3. 5), then cp is weakly nuclear.

Proof. The only issue is how to replace a net of maps which have norms
wandering off to infinity with contractions. So let 'Pn: A ------ Mk(n) ((['.) and
'I/Jn: Mk(n)(C)------ M be c.p. maps whose compositions converge in the point-
strong operator topology to cp. (Since ultraweak convergence implies weak
convergence, we have used the previous lemma to get a better topology.)
As we have seen many times, Lemma 2.2.5 allows us to assume that 'Pn is
unital for every n.
Since both cp and the 'Pn's are unital, it follows that 'l/Jn(l) ------ 1 in the
strong operator topology (though the norms of 'l/Jn(l) may be large). Now
fix o > 0 and let Pn EM be the spectral projection of 'l/Jn(l) corresponding
to the interval [O, 1 + o]. Since llPnll = 1, Pn('l/Jn o 'Pn(x) - cp(x))------ 0 in the
strong operator topology, for every x EA. Since l'l/Jn(l)-11 ;:::: oP,;/:, we have
that Pn------ 1 in the strong operator topology. Hence, Pn('l/Jno'Pn(x))------ cp(x)
in the strong operator topology as well.