1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.3. Nuclear and exact C*-algebras 35

Remark 2.3. 7. The proof shows that the set of factorable maps which have

contractive factorizations (i.e., () = 'lj;o<p where both <p and 'ljJ are contractive)

is also convex.

Proposition 2.3.8. If A** is semidiscrete 1 then A is nuclear.^6

Proof. Assume first that A is unital and fix a finite set .;y c A, a finite set
x c A* = (A*) and c: > 0. By the previous lemmas, it suffices to show that
idA belongs to the point-weak closure of the contractive factorable maps -
i.e., there exist c.c. p. maps <p: A ----+ Mn ( q and 'ljJ: Mn ( q ----+ A such that
1"7('l/J o r.p(a)) - 'Tl(a)I < E
for all a E .;y and 'Tl E X·
Since A is semidiscrete, we can apply Proposition 2.2.7 to find u.c.p.
maps <p: A ----+ Mn(C) and 'lf;': Mn(C) ----+A such that 1"7('l/J' o r.p(a)) -
'Tl(a)I < E for all a E .;y and 'Tl EX· Of course, if 'lf;' happens to take values in
A, then we are done - so this is what we'll arrange.
Using the duality between c.p. maps Mn(C) ----+ A and positive ele-
ments in Mn(A) (Proposition 1.5.12) and the fact that positive elements
in Mn(A) are ultraweakly dense in the positive part of Mn(A), it is not
hard to find a net of c.p. maps 'l/J>..: Mn(C) ----+ A such that 'l/J>.. ----+ 'lj;' in the
point-ultraweak topology. Unfortunately the 'l/J>..'S need not be contractive.
However, since <p and 'lj;' are unital maps, we do have that { 'l/J>.. (lMn(C))} C A
is a net converging weakly to lA. Hence, going far enough out in the net
and taking an appropriate convex combination, we can find a c.p. map
'lj;": Mn(C) ----+A such that for all a E .;y and 'T] Ex,

1"7('l/J" o r.p(a)) - 'Tl(a)I < c:

and ll'l/J"(lMn(q) - lAll < c:. To get a contractive map, we now replace 'lf;"

by

and one easily verifies that 'lj;o<p is close to the identity on .;y for the prescribed
finite set of functionals.

In the nonunital case one first shows that a C* -algebra is nuclear if and
only if its unitization is nuclear (cf. Exercise 2.3.4). Then one observes
that (.A) = A EB C (in general, if J <l B is an ideal, then B ~ J EB
(B / J)), which easily implies that (.A) is semidiscrete whenever A** is
semi discrete. D

Here are a few more useful observations.

6The converse also holds, but it requires the equivalence of injectivity and semidiscreteness;

see Theorem 9.3.3.