1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.5. Compact group actions and graph C*-algebras 133

Exercise 4.4.3. Prove that if Y is a compact r-space with a r-equivariant

continuous map f: Y-+ X, then the action of r on Y is also amenable.

Exercise 4.4.4. Show that if X admits a r-invariant probability measure,

then r itself is amenable. (Hint for C* -enthusiasts: The nuclear C* -algebra

C(X) ><l r has a tracial state in this case.)

4.5. Compact group actions and graph C*-algebras

In this section we will give a simple proof of the fact that the C* -algebras
arising from directed graphs are always nuclear. Our goal here is not to study
these important examples properly; we want to introduce a new method of
proving nuclearity in a specific context (setting the stage for the next section
where the same technique will be used for Cuntz-Pimsner algebras). We will
try to reach nuclearity as quickly as possible and hence quote a few results
without proof. The interested reader can consult [163] for the things we
omit.

So far we have only considered actions of discrete groups on C -algebras,
but there are plenty of important examples of nondiscrete group actions.
The main idea we wish to advertise is that when a C -algebra admits an ac-
tion of a compact group, the question of nuclearity (or exactness) is reduced
to understanding the fixed point algebra.

If K is a locally compact group, then an action of K on A is a homo-
morphism a: K-+ Aut(A) which is continuous in the point-norm topology
(i.e., g 1-+ ag(a) is a con~inuous map, from the given topology on K to the
norm topology on A, for every a E A). The fixed point subalgebra A°' is
defined to be the set of a E A such that ag (a) = a for all g E K. If K hap-
pens to be a compact group, then there is always a conditional expectation
Ea: A -+ A°' given by

Ea(a) = l az(a) dz,

where the integration is with respect to Haar measure on K. We note that
Ea is faithful. Indeed, let a E A+ \ {O} and choose a state <p on A with
<p(a) > 0. Then, the function K 3 z 1-+ <p(az(a)) is nonzero and nonnegative.
Thus, <p(Ea(a)) = JK <p(az(a)) dz> 0, proving Ea(a) -I 0.
The following fact is well known, having been exploited by Cuntz and
many others.

Proposition 4.5.1. Let A and B be C-algebras, a and (3 be actions of a
compact group K on A and B 1 respectively, and 7r: A -+ B be an equivariant
-homomorphism. Then, 7r is injective if and only if it's injective on the fixed
point algebra A°'.