1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

116 4. Constructions

It turns out that nuclearity and exactness are reasonably well behaved

under this construction and, in most cases, one can construct explicit finite-

dimensional approximations on the crossed product. As such, we'll give a

detailed description of these important examples, trying to stay as concrete

as possible.

Definition 4.1.1. Let r be a discrete group and A be a C*-algebra. An

action of r on A is a group homomorphism a. from r into the group of

*-automorphisms on A. A C*-algebra equipped with a I'-action is called a

I' -C* -algebra.^1

Our goal is to construct a single C* -algebra which encodes the action

of r on A. In group theory, this procedure is well known and is called the

semidirect product. We will adapt this idea and create an algebra A ><I°' r

with the property that there is a copy of r inside the unitary group of A ><1 °' r

(at least when A is unital) and there is a natural inclusion Ac A ><la r such

that (a) A ><1 °' I' is generated by A and I' and (b) a 9 (a) = gag* for all a E A

and g EI' (i.e., the action of I' becomes inner).

For a I'-C*-algebra A, we denote by Cc(I',A) the linear space of finitely

supported functions on r with values in A. A typical element Sin Cc(I', A) is

written as a finite sum S = L:sEI' ass· We equip Cc(I', A) with an a-twisted

convolution product and *-operation as follows: for S = L:sEI' ass, T

I:tEr btt E Cc(r, A) we declare

SaT= L asas(bt)st and S = I::as-1(a:)s-^1.
s~EI' sEI'
The twisted convolution is a generalization of the classical convolution of two
£^2 (Z) functions, but the algebraic explanation of these formulas is perhaps
more enlightening. Indeed, we are trying to turn Cc(I',A) into a *-algebra
where the action becomes inner and hence the definition above comes from
the formal calculation

(~= ass)(L btt) = L as(sbts*)st = L asas(bt)st.

sEI' tEI' s,tEI' s,tEI'

However you care to think about it, Cc(I', A) is the smallest -algebra which
encodes the action of r on A. Note that when A= C and the action a is
trivial, we simply recover the group ring C[r]. Now the question is, "How
shall we complete Cc(I', A)?" Just as for group C-algebras, there are two
natural choices, a universal and a reduced completion. ·
A covariant representation (u, 7r, 1-i) of the I'-C-algebra A consists of a
unitary representation ( u, 1-i) of r and a -representation ( 7r, 1-i) of A such

(^1) 0f course, there could be many different actions of r on A, giving rise to different I'-C*-
algebra structures.