1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

124 4. Constructions

Corollary 4.2.5. The rotation algebras Ae are nuclear.

At this point, the reader may feel lied to - we said we would stick to

Z-actions in order to stay concrete and then proceeded to keep !Z out of the

picture until the very end. Here is a corollary of our deceit (the proof is

identical to the proof of Theorem 4.2.4).

Theorem 4.2.6. For any amenable group r and action a: r ----+ Aut(A),

the following statements hold:

(1) A><laI'=A><la,rI';

(2) A is nuclear if only if A ><la r is nuclear;

(3) A is exact if and only if A ><la r is exact.

4.3. Amenable actions

We now step up the generality ladder and consider crossed products by
amenable actions - i.e., the group involved need not be amenable, but we
require it to act nicely. When defined "appropriately" (not the definition
usually found in the literature, but an equivalent one that makes our present
work easier) and done abstractly, finding approximating maps on a crossed
product by an amenable action is only slightly harder than the case of !Z
actions.
Given a I'-C-algebra A, we put a third norm on the (a-twisted) con-
volution algebra Cc (I', .fl): for finitely supported functions S, T: r ----+ A we
define
(S, T) = L S(g)T(g) EA
and
llSll2 = ll(S, 8)11^112.
The informed reader will notice that we have made a Hilbert C -module -
more on that subject in Section 4.6. The Cauchy-Schwarz inequality holds
in this context: ll(S,T)llA::::; llSll2llTll2, for all S,T E Cc(I',A).^3
Definition 4.3.1. An action a: r----+ Aut(A) on a unital C-algebra A is
amenable if there exist finitely supported functions Ti : r ----+ A with the
following properties:

(1) 0::::; Ti(g) E Z(A) (the center of A) for all i EN and g Er;

(2) (Ti, Ti)= 'L: 9 ErTi(g)^2 = lA;

(^3) This is a general fact about Hilbert modules, but here we only need the case that A is
abelian. If A= C(X), the asserted inequality follows from the usual Cauchy-Schwarz inequality,
applied pointwise in X.