1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

132 4. Constructions

Remark 4.4.4. This theorem provides natural examples of subalgebras of

nuclear C*-algebras which are not nuclear. Indeed, if r is any nonamenable

group which admits an amenable action on some compact space X, then

C~(r) c C(X) ><l r is a nonnuclear subalgebra (see Theorem 2.6.8). Such

groups abound, as we will see in the next chapter.

We close this section with a few non-C* -characterizations of amenable

actions on compact spaces.

Proposition 4.4.5. Let X be a compact topological I'-space. The following

are equivalent:

(1) the action is amenable;

(2) for any finite subset E c r and c > 0, there exists a continuous

map~: X-+ £^2 (r) such that ll~xll2 = 1 for all x EX and

max sup lls.~x -~s.xll2 < c;

sEE xEX

(3) for any finite subset E c r and c > 0, there exist a finite subset

F C r and a family of nonnegative continuous functions Ut)tEF on

X such that L,tEF ff = 1 and

max sup 11- ~ ft(s-^1 .x)fst(x)I < c.

sEE x EX tEFns-~ (^1) F
Proof. The assertion (1) 9 (2) follows from the fact that μ f-+ μ^112 is a
uniform homeomorphism from Prob(I') into £^2 (r) (see the estimates used in
the proof of Lemma 4.3.7). For (2) 9 (3), we define ft(x) = ~x(t) (and vice
versa). Then for every x EX, t f-+ ft(x) defines a unit vector in £^2 (r) and
a calculation shows that
max sup 11- L ft(s-^1 .x)fst(x)I =max sup ll - (s-^1 .~x, ~s-1.x)I,
sEE xEX tEFns- (^1) F sEE xEX
where the set F comes from Lemma 4.3.8. Since unit vectors in Hilbert
space are close in norm if and only if their inner product is almost one, the
proof is complete. D
Exercises
Exercise 4.4.1. Prove that any action of an amenable group on a compact
space is amenable. Prove that the trivial action of r on a one-point set is
amenable if and only if r is amenable.
For the next three exercises, assume that r acts amenably on X.
Exercise 4.4.2. If I'o is a subgroup of r, then r 0 also acts amenably on X.