1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

9.4. Corollaries 297

Proof of Theorem 9.3.3. Let M be an injective von Neumann algebra

and a be a modular action so that M ><la IR is semifinite (Theorem 9.3.5).

Since M ®lB(L^2 (IR)) is injective (which isn't as obvious as you might think)

and M ><la IR is the range of a conditional expectation from M ® JB(L^2 (IR))

by Theorem 9.3.7 and Lemma 9.3.6, M ><la IR is also injective. It follows

from Theorem 9.3.4 that M ><la IR is semidiscrete. Let { 'lj;i} be a net of

finite-rank u.c.p. maps on M ><la IR which converges to the identity in the

point-ultraweak topology. By Lemma 9.3.6, there exists a net of normal

u.c.p. maps 'Pn: M ><la IR ____., M which converges to the identity on M. It

follows that 'Pn o 'lf;ilM are finite-rank u.c.p. maps on M such that

ultraweak-limlim n i 'Pn o 'lj;i(a) = ultraweak-lim n 'Pn(a) =a

for every a E M. This proves semi discreteness of M. D

Exercise
Exercise 9.3.1. Prove that A has property C if and only if A has both
properties C' and C". Use this fact, and Lemma 9.2.8, to remove the sepa-
rability hypothesis in Theorem 9.3.1.

9.4. Corollaries

Corollary 9.4.1. Exact C* -algebras are locally reflexive.

Proof. Assume A is separable and exact. Theorem 9.3.1 implies A has
property C; evidently this implies property C". Hence, by Proposition 9.2.5,
A is locally reflexive. The nonseparable case follows, thanks to Lemma
9.2.8. D

Remark 9.4.2. It follows that quotient maps of exact C* -algebras are al-
ways locally liftable (Proposition 9.1.4).

Corollary 9.4.3. Quotients of exact C* -algebras are exact.

Proof. Since an algebra is exact if and only if all its separable subalge-
bras are exact, we may assume separability. Now apply Theorem 9.3.1 and
Proposition 9.2.4. D

Corollary 9.4.4. Quotients of nuclear C* -algebras are nuclear.

Proof. Let A be nuclear and J <l A be an ideal. It suffices to show (A/ J)
is semidiscrete (Proposition 2.3.8). But A = J EB (A/ J), so it suffices
to show A** is semidiscrete (this clearly passes to direct summands). But
we outlined this fact in the paragraph preceding Theorem 9.3.3. D

Combining with a deep result of Glimm, we recover a theorem of Black-
adar ( [1 7]).