1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

130 4. Constructions

Definition 4.4.1. We say a function h: Xx r ~ <C is of positive type if for

any finite sequence s1, ... , Sn Er and x EX, the matrix [h(si.X, Sisj^1 )]i,j E

Mn(<C) is positive.

Since an element TE Mn(C(X)) = C(X, Mn(<C)) is positive if and only

if the matrix T(x) is positive for all points x E X, Corollary 4.1.6 implies

that a function h E Cc(X x r) is of positive type if and only if it is positive

when regarded as an element in Cc(r, C(X)) c C(X) ~a,r r.

Definition 4.4.2. Let X be a compact topological I'-space. A I'-C*-algebra

A is called a ( X ~ I')-C* -algebra if C ( X) is embedded in the center of A in

such a way that a 8 (C(X)) = C(X) and a 8 (f)(x) = f(s-^1 .x) for every s Er,

f E C(X) and x EX. In other words, there is an equivariant embedding of

C ( X) into the center of A.

Theorem 4.4.3. Let X be a compact topological I'-space. The following are

equivalent:

(1) the action of r on X is amenable;

(2) for any finite subset F c r and c > 0, there exists a positive type

function h E Cc(X x I') such that

max sup lh(x,s)-11 < s;

sEF xEX

(3) for any (X ~ I')-C*-algebra A, we have A~°' r =A ~a,r I';

(4) the C*-algebra C(X) ~a,r r is nuclear.

Proof. (2) =? (1): Let h E Cc(X x r) be a positive-type function as in
condition (2). Since his positive in the C-algebra C(X) ~a,rI', there exists
g E Cc(X x r) such that Ilg a g - hll < E, where II. II is the C-norm on
C(X) ~a,r I'. Let E: C(X) ~a,r r ~ C(X) be the standard conditional
expectation. Since E(g a g) ~ E(h) ~ 1 in C(X), replacing g with g a
E(g a g)-^1 /^2 (and changing E > 0), we may assume that E(g *a g) = 1.
We define T: I'~ C(X) by T(t)(x) = lg(r^1 .x, r^1 )1. Then,

\T, s *a T)(x) = (LT(t)a 8 (T(s-^1 t))) (x)