4.2. Integer actions 121When proving that a particular map into a crossed product is completely
positive, it often suffices to prove positivity. We will use the following exer-
cise (taking B to be a matrix algebra) for this reduction.Exercise 4.1.3. If a: r----" Aut(A) is an action and T®a: r----" Aut(B®A)
is defined by (T ® a) 9 = idB ® a 9 , then
(B ®A) ><1 7 @a:,r r ~ B ®(A ><lo:,r r).
What is the corresponding result for universal crossed products?
Exercise 4.1.4. Let (A, a) and ( B, /3) be r-C -algebras and <p: A ----" B be
a c.c.p. map which is r-equivariant. Prove that the map rp: Cc(r, A) ----"*
Cc(r, B), defined by <P(I::s a 8 s) = L:s <p(as)s, extends to a c.c.p. map from
A ><10:,r r into B '><1f3,r r (resp. from A '><lo: r into B '><1f3 r). (Hint: The reduced
case is easy. For the full case, you'll need a r-equivariant Stinespring Dilation
Theorem.)
4.2; Integer actions
We now specialize to the case of Z actions, where things are easier to di-
gest. Our goal is to construct explicit approximating maps on the crossed
product. This approach is a little boorish, but it has been very impor-
tant for other purposes (e.g. noncommutative entropy theory or calculating
Haagerup invariants).
Suppose A c lffi('h'.) and a E Aut(A) is an automorphism. We also
use a to denote the induced action of Z given by n 1-+ an. Let [k, n] =
{ k, k + 1, k + 2, ... , n} be the interval of integers from k to n. The finite
subsets Fn = [O, n] will play.an important role as they happen to be a F¢lner
sequence: for each k E Z,
J(k+Fn)nFnl = J[k,n+k]n[O,n]J = n-k+l--*l,
IFnl n+ 1 n+ 1
as n ----"* oo, where J · J denotes cardinality. It turns out that we can easily
construct approximating maps by cutting to F¢lner sets and then mapping
back to the crossed product. We will need a few simple lemmas.
Lemma 4.2.1. Let A be a C* -algebra and let n E N. Every positive element
in Mn(A) is a sum of n elements of the form [aiaj]f.j=I · '
Proof. Take an arbitrary positive element x E Mn(A) and decompose it as
a product x = [bij]*[bij]· Now one writes
[bij] = A1 + A2 + · · · + Anand