134 4. ConstructionsProof. If a EA+ and 7r(a) = 0, then 7r(Ea(a)) = E13(7r(a)) = 0. D
Theorem 4.5.2. Let A be a C*-algebra and a be an action of a compact
group K on A. Then, A is nuclear (resp. exact) if and only if A°' is nuclear
(resp. exact).Proof. Since A°' is the range of a conditional expectation, the "only if'
parts are easy.
Let's show that nuclearity of A°' implies the same for A. For an arbitrary
B, we can tensor with the trivial action a @ id B to get actions of K on both
A @max B and A @ B. Since A°' is the range of a conditional expectation,
there is a canonical inclusion A°' @max B C A @max B (Proposition 3.6.2).
The key observation is that
(A @max B)°'®idB =A°' @max B,
since the conditional expectation evidently maps elementary tensors into
A°' @max B (and linearity and continuity forces everything else into the same
algebra). The same argument works for minimal tensor products and hence
we have the commutative diagram
A@maxB -----+ A@B~
A°'@maxB -----+A°' @B,
where the bottom row is an isomorphism of fixed point algebras. Thus the
previous proposition implies injectivity of the top row, so we see that A
must be nuclear.
The case that A°' is exact uses a similar argument. For arbitrary Band
ideal J <l B we consider the commutative diagram
A©B
A©Jr
A@ (B/J)
r
A"'©B A"'©J -----+ ~ A°'@ ( B I J. )
One must again show that the bottom row consists of the fixed point sub-
algebras of the appropriate actions - recall that there is an embedding
A 8 (B / J) C 1~~ and this dense subalgebra evidently gets pushed into
A°' 8 (B / J) C 1:~~ under the conditional expectation onto the fixed point
subalgebra - and conclude the proof as above. D
We now apply this theorem to the class of C* -algebras arising from
directed graphs. By definition, a directed graph <B = (V, E, s, r) consists
of a set V of vertices, a set E of edges and two maps s, r: E ---+ V, called
the source and range maps (s(e) is the vertex at which an edge e begins,