1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

9.3. Equivalence of exactness and property C 293

Proof. Let <p: E _,A** be a complete contraction with dim(E) < oo. By

operator space duality, <p corresponds to a norm-one element z E E* ® A**.

Our task is to show JJzil(E*@A)** ::;: 1. (Note that since dim(E) < oo, we

have a canonical algebraic identification E* ®A**= (E* ®A)**.)

Take a universal representation A** C llll(H) and let B = A+IK(H). We

regard <p as a map into B" C B**. Clearly, we can approximate <p by a net

of complete contractions 'Pi: E _, B in the point-ultraweak topology. In

other words, we have JJzii(E*@B)** ::;: 1. Since the embedding (E* ®A)** c

(E* ® B)** is isometric, we deduce that JJzii(E*@A)** ::;: 1. D

9.3. Equivalence of exactness and property C

This section contains the main result. We still have a lot of von Neumann
algebra work to do, but after that only a few simple remarks remain. Why
not proceed in reverse? The main theorem is
Theorem 9.3.1. For a separable C -algebra A, the following are equivalent:
(1) A is exact;
(2) A has property C;
(3) A has property C'.
Since property C evidently implies property C", it follows that every exact
C -algebra is locally reflexive.

Proof. Evidently property C implies property C'. Also, the equivalence of
conditions (1) and (3) is the content of Proposition 9.2.7. Thus we must
show (1) =? (2).
However, if A is exact, then it is isomorphic to a subquotient of a nuclear
C-algebra (Corollary 8.2.5). If we knew that every nuclear C-algebra had
property C, then we could appeal to Propositions 9.2.3 and 9.2.4 to complete
the proof. Thus we devote the remainder of this section to proving nuclear
C*-algebras have property C. D

There are two steps involved. The first is very easy.

Proposition 9.3.2. If A** is semidiscrete, then A has property C.

Proof. Since A is semidiscrete, the map A, A C (A® B) c llll(H)
is weakly nuclear (into A) for any Band normal representation (A@B) c
llll(H). Theorem 3.8.5 then implies that A 8 B , (A® B)** c llll(H) is
min-continuous as desired. D

The second step is to show that the double dual of a nuclear C* -algebra
is semidiscrete. However, a von Neumann algebra is semidiscrete if and only