26 2. Nuclear and Exact C* -AlgebrasNote that nuclear maps are automatically c.c.p. If A, B and e are
unital, we'll soon see that the 'Pn's and 'l/Jn's can be replaced with u.c.p.
maps (Proposition 2.2.6).
Though they won't get used as much as the C*-version, we will need
nuclear maps in the von Neumann algebra context too.
Definition 2.1.2. If A is a C*-algebra and N is a von Neumann algebra, a
map e: A ------> N is called weakly nuclear if there exist c.c.p. maps !.pri: A ------>
Mk( n) ( <C) and 'I/Jn : Mk( n) ( <C) ------> N such that 'I/Jn o r.pn ------> B in the point-
ultraweak topology:
ry('l/Jn o r.pn(a))------> ry(B(a)),
for all a E A and all normal functionals 'T/ E N*.As in the C* -context, when e is a weakly nuclear unital map, we can
replace c.c.p. approximations by u.c.p. maps; we can also arrange normality
(whether or not e is normal). See Propositions 2.2.7 and 2.2.8 in the next
section.
Remark 2.1.3. It follows from Sakai's predual uniqueness theorem that
when checking point-ultraweak convergence of bounded nets, it always suf-
fices to check convergence on certain vector functionals. That is, if N C
IIB(H) is any faithful normal representation and n c 1i is any set of vectors
whose linear span is dense in H, then 'I/Jn o r.pn ------> e in the point-ultraweak
topology if and only if
('I/Jn o r.pn(a)v, w) ------> (B(a)v, w)
for all a E A and v, w E D. If the set D is a linear subspace, then the
polarization identity implies that one only need check the positive vector
functionals arising from n.A fundamental subtlety which makes nuclear maps irritating and/ or
interesting is the dependence on the range. More precisely, it often happens
that a map e: A ------> B is not nuclear, but after embedding B into some larger
algebra 0, it becomes nuclear. In fact, this is the difference between exact
and nuclear C* -algebras.
We will have to wait for C* -examples, but this phenomenon is readily
seen in the von Neumann algebra context. Indeed, there are many con-
crete examples of von Neumann algebras MC IIB(H) for which the identity
map idM: M------> Mis not weakly nuclear (cf. Theorem 2.6.8); however, the
natural inclusion M "---+ IIB(H) is always weakly nuclear.Proposition 2.1.4. Let MC IIB(H) be a von Neumann algebra. The natural
inclusion map M "---+ IIB(H) is always weakly nuclear.