1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

126 4. Constructions

Define
X = L a;^1 (T(p)) ® ep,p
pEF
and note that X = X*. Hence compression by X is a c.p. map and a
computation confirms that

X<p(a>. 8 )X = L a;^1 (T(p)a)a~! 1 P(T(s-^1 p)) ® ep,s-lp·

pEFnsF

We know the map A® Mp(C) -t A ><la,r r defined by

b® ex,y ~ ax(b)Axy-1

is u.c.p., so we get another u.c.p. map 'l/;: A® Mp(C) -t A ><la,r r by com-
posing it with compression by X:

X X b®ex yi--+ax(b).>..xy-1

'lj;:A®Mp(C) ~ A®Mp(C) '-+ A><1a,rr.

Finally, since T(g) E Z(A) for all g Er, we have

'l/; o 1.p(a>. 8 ) = L ap( a;^1 (T(p)a)a,;_\(T(s-^1 p)) )>.s

pEFnsF

= ( L T(p)a 8 (T(s-^1 p)) )a>. 8

pEFnsF

= (T *a T*(s))a>.s.

D

The proof of the next theorem is but a tiny perturbation of that given

for Theorem 4.2.4. We leave the details to the reader.

Theorem 4.3.4. For any amenable action of r on A, the following state-

ments hold:

(1) A ><la r =A ><la,rr;

(2) A is nuclear if only if A Xia r is nuclear;

(3) A is exact if and only if A Xia r is exact.

We call a compact^5 space X a r-space if it is equipped with an ac-

tion of r (by homeomorphisms). Let x ~ s .x denote the action of s E r

on x E X. To help distinguish, we let a 8 : C(X) -t C(X) denote the in-

duced automorphism of C(X) (i.e., a 8 (f)(x) = f(s-^1 .x)). The notion of an

amenable action comes from classical (i.e., abelian) dynamical systems. As

already mentioned, our definition at the C*-level is not very common in the

literature. Here is a more popular version.

(^5) As usual, compactness includes the Hausdorff axiom.