4.2. Integer actions 123By Lemma 4.2.1, we only need to check that for every set {ap}pEF c A,
'I/;(~ a;aq ® ep,q) 2:: 0. But'I/;( L a;aq ® ep,q) = L [~[ cx:p(a;aq)>.pq-1,
p,qEF p,qEF
so the previous lemma completes the proof. 0
Theorem 4.2.4. For an automorphism ex: E Aut(A), the following state-
ments are true:
(1) A ><la Z =A ><la,r Z;
(2) A is nuclear if and only if A ><la Z is nuclear;
(3) A is exact if and only if A ><la Z is exact.Proof. Proof of (1): It suffices to show that there exist c.c.p. maps
Wn: A ><lar Z--+ A ><la Z
'
such that \x - Wn o 7r(x)\u----+ 0 for all x E Cc(Z, A) CA ><la Z, where
7r: A ><la Z----+ A ><lar Z
'
is the canonical quotient map (coming from universality).
The key observation is that the proof of Lemma 4.2.3 is algebraic. In
other words, if Fn = [O, n] C Z is a F¢lner sequence and <pn, 'l/Jn are the
corresponding maps constructed in Lemma 4.2.3, then we can define c.c.p.
maps Wn: A ><la,r Z ----+ A ><la Z by Wn = 'l/Jn o i.pn, but simply regarding the
'l/Jn's as taking values in the universal crossed product (as opposed to the
reduced one, since the range of 'l/Jn is contained in Cc(Z, A)). The formula
in Lemma 4.2.3 still holds, and hence for x = l:kEZ akk E Cc(Z, A) we have
\\x - '1tn(7r(x)) \\A)qaZ = II L(l - IFn n I~ T Fn)l )akk[[A)qaZ ----+ 0
kEZ nsince only finitely many ak 's are nonzero.
Proofs of (2) and (3): Both of the "if'' directions are trivial since there
is a conditional expectation A ><la Z----+ A.
For the other direction one should first review Exercises 2.3.11 and
2.3.12. Indeed, another way of stating Lemma 4.2.3 is that there exist c.c.p.
maps
that 'l/Jno'Pn----+ id in the point-norm topology. Since A®Mk(n)(C) is nuclear
(resp. exact) whenever A is nuclear (resp. exact), we are done. 0
Recall that for a real number e > 0, the rotation algebra Ae is defined
to be C('TI') ><lae Z where cx:e is the automorphism induced by a rotation of
the circle through an angle of 27r0 radians.