1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

38 2. Nuclear and Exact C*-Algebras

Exercise 2.3.13. The proof of Proposition 2.3.8 goes through verbatim if

one only assumes that the natural inclusion A "4 A** is weakly nuclear. Do

you agree?

Exercise 2.3.14 (Lance's WEP). A C*-algebra A is said to have the weak

expectation property (WEP) if there exists a u.c.p. map <J?: JIB(Hu) ---+ A**

such that <l?(a) =a for all a EA, where Ac A** c JIB(Hu) is the universal

representation of A. Prove that A is nuclear if and only if A is exact and

has the WEP. (Hint: Use Arveson's Extension Theorem and point-ultraweak

limits - Theorem 1.3. 7 - to prove that nuclearity implies the WEP.)

Prove also that A has the WEP if and only if for any faithful representa-

tion Ac JIB(H), there exists a u.c.p. map <J?: JIB(H) ---+A" such that <J?(a) =a,

for all a EA.

Exercise 2.3.15 (Semidiscrete implies injective). Prove that every semidis-

crete von Neumann algebra is injective.

2.4. First examples

It turns out that most algebras built out of abelian and finite-dimensional

C*-algebras will be nuclear. In K-theoretic terms, this already provides a

huge class of examples. On the other hand, von Neumann algebras provide

nice examples of nonexact (hence nonnuclear) algebras, so we discuss this

as well.

Exercise 2.1.2 obviously implies that every finite-dimensional C* -algebra

is nuclear. Since inductive limits of nuclear algebras are nuclear (Exercise

2.3.7), we have the following fact.

Proposition. 2.4.1. Approximately finite-dimensional (AF) algebras are

nuclear.

Proof. By definition, AF algebras are inductive limits of finite-dimensional

C* -algebras. D

A slightly less trivial, though equally fundamental, example is that of

abelian algebras.

Proposition 2.4.2. Every abelian C* -algebra is nuclear.

Proof. It suffices to prove this in the unital case - i.e., we may assume that
A= C(X) for some compact Hausdorff space X. Despite what topologists
say, partitions of unity were invented specifically to prove this proposition.

If a finite set :;y c A and E > 0 are given, then we can find a finite open
cover {U1, ... , Un} of X with the property that for each f E :;y and 1 ::=; i ::=; n