4.4. x )<I r -algebras 129
The proof of this lemma shows that if X is an amenable r-space, then
we can assume each map mi: X --+ Prob(r) has the property that there
exists a finite set Fi c r with supp(mf) c Fi, for every x E X. Here is a
direct proof of this fact.
Lemma 4.3.8. Let m: X --+ Prob(r) be a continuous map. Then, for any
c > 0, there exist m: X --+ Prob(I') and a finite subset F c r such that
suppmx CF and llmx - mxll1 < c for all x EX.Proof. For every finite subset F c r, let U(F) = {x E X : llmxxFll1 >
1 - c/2} C X, where XF is the characteristic function of F. It is easily
seen that {U(F)}p is an open cover of X which is upward directed. Since
X is compact, there exists F such that X = U(F). It follows that mx =
mxXF + llmxXr\Fll16e has the desired property. D
Exercises
Exercise 4.3.1. Assume r x r acts on C(X) and there exist two r x r-
invariant subalgebras A, B c C(X) such that (a) r x { e }IA is amenable while
{e} x I'IA is trivial and (b) I' x {e}iB is trivial while {e} x I'iB is amenable.
Prove that the action of r x r on C(X) is amenable. (In addition to helping
cement Definition 4.3.5 in your mind, this exercise will be needed later; see
Corollary 5.3.19.)
Exercise 4.3.2. Find an elementary proof, based on Fell's absorbtion prin-
ciple, of the fact that A ><1 r = A ><Irr, whenever the action is amenable.
(Hint: Fix an embedding A ><1 r c IIB('h'.) and use the definition of amenable
action to construct some isometries from 1-{ to 1-{ Q9 £^2 (r).)
4.4. x )<I r-algebras
The previous section says precious little about when an action is amenable.
The point was that if it is, then crossed products are well behaved. It turns
out that determining amenability is inextricably intertwined with nuclearity
(and all tangled up with exactness too). This section is devoted to exposing
this connection, in a slightly more general context.
Instead of considering the *-algebra of finitely supported functions from
r to C(X), it is often convenient to think of compactly supported functions
Xx r --+ C. That is, since any compact subset of Xx r is contained in Xx F,
for some finite subset F c r, we can identify Cc(r, C(X)) with Cc(X x r)
- an element l::::sEr fss E Cc(r, C(X)) corresponds to f E Cc(X x r), where
f(x, s) = f 8 (x). One checks that our a-twisted convolution and adjoint on
Cc(I', C(X)), when transferred to Cc(X x r), look like
(g a f)(x, s) = Lg(x, t)f(r^1 .x, r^1 s) and f(x, s) = f(s-^1 .x, s-^1 ).
tEr