1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

11.2. NF and strong NF algebras 319

commutative diagram

Ai -------> Ai EB A2 -------> Ai EB A2 EB A3 ------->... -------> B

1 1 1 1

Ai -------> A2 -------> A3 ------->... -------> A.

Since nuclearity passes to quotients, this implies that A is nuclear. That

nuclear MF algebras are QD follows easily from the Choi-Effros Lifting The-

orem (Theorem C.3) and Exercise 7.1.3. D

Now let's prove the converse.

Theorem 11.2.5. For a separable C* -algebra A, the following statements

are equivalent:

(1) A is an NF algebra;

(2) A is nuclear and isomorphic to a subalgebra of

AC ITnEN Ms(n) (CC)

E9nEN Ms(n) (CC)

for some sequence s(n);

(3) A is nuclear and QD;

( 4) for every finite set:;$ C A and c > 0 there exist c. c. p. maps a: A ---+

Mn(CC) and (3: Mn(CC) ---+ A such that 11/3 o a(a) - ail < c and

lla(ab) - a(a)a(b)ll < 01 for all a,b E :;$. {In other words, the

definitions of nuclearity and quasidiagonality can be incorporated

into the same c.c.p. maps.)

Proof. Thanks to the previous proposition, we only have to prove (3) ==?-(4)
and (4) ==?-(1).

Assume A c lffi('h'.) is nuclear, QD and that it contains no nonzero
compact operators. Fixing :;$ c A and 0 > 0, we can apply Dadarlat's
approximation theorem (Theorem 7.5.7) to find a finite-dimensional alge-
bra B c lffi('h'.) which nearly contains :;$. We can also find c.c.p. maps
a: A --+ Mn(<C) and f3: Mn(<C) --+ A such that llf3 o a(a) - all < 0 for all
a E :;$. By Arveson's Extension Theorem we may assume that a is defined
on all of lffi('h'.). Let 1.p: lffi('h'.) --+ B be a conditional expectation. One easily
checks that a= lflA: A--+ Band f3 = 'fJ o aiB: B--+ A are the desired maps,
thus proving ( 4).

Now assume the approximation property in statement (4) and we will
construct the right generalized inductive system. First, find some finite-
dimensional self-adjoint subspaces

Sic S2 c · · · c A