1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

34 2. Nuclear and Exact C*-Algebras

We must show that for. each finite set J = { ai, ... , ak} C A and E > 0

there exists SEC such that llS(aj) -T(aj)ll < E for 1 :S j :S k. This is a

standard application of the Hahn-Banach Theorem since the net of maps

Ti E9 · · · E9 Ti (/>:-fold direct sum)

converges point-weakly to

TE9···E9T

in the Banach space B(AEEl· · ·EElA) (take your favorite £P-norm on AE9· · ·EElA).

Thus the element

T( ai) E9 · · · E9 T( ak)

belongs to the weak closure of

{7i(a1) E9 · · · E9 7i(ak)hEI

and hence, by the Hahn-Banach Theorem, also to the norm closure of the

convex hull. This implies that we can find a single convex combination of

the Ti's which is simultaneously close (in norm) to Ton all of J. D

The second lemma is awkward to state if we don't introduce some ter-

minology.

Definition 2.3.5. We will say a c.p. map e: A --+ A is factorable if there

exist c.p. maps cp: A--+ Mn(q and 7/J: Mn(C) --+A such that e = 'ljJ o cp.

We call (cp,7/J,Mn(C)) a factorization of e.

Note the absence of restrictions on the norms of cp and 'ljJ (which will be

handy later).

Lemma 2.3.6. For any C* -algebra A the set of factorable maps A --+ A is

convex.^5

Proof. Let B1, B2 : A --+ A be factorable maps and 0 < >. < 1 be given. If
factorizations ( cpi, 7/Ji, Mn(i) (CC)) of ei are given, then we claim that >.B 1 +
(1->.)B2 factorizes through the finite-dimensional C*-algebra
Mn(l)(CC) E9 Mn(2)(C).
Indeed, factorizations are given by
I.pl E9 cp2 : A --+ Mn(l) ( q E9 Mn(2) ( q
and Mn(l)(C) E9 Mn(2)(C)--+ A,
T E9 Sr--+ A7/J1(T) + (1->.)7/J2(S).
Since the set of positive operators is a cone, this last map is c.p. Now one
completes the proof as in Exercise 2.1.2. D

(^5) 0ne could also consider factorable maps with values in a different C* -algebra, but not much
changes.