4.1. Crossed products 117that Us1f(a)u; = 7r(a 8 (a)) for every s E r and a E A. rt· is not hard
to see that every covariant representation gives rise to a -representation of
Cc (r, A) and, conversely, every ( nondegenerate) -representation of Cc (r, A)
arises this way. For a covariant representation ( u, 7r, H), we denote by u x 7r
the associated -representation of Cc(I',A).
Definition 4.1.2. The full crossed product (sometimes called the "univer-
sal" crossed product) of a C-dynamical system (A, a, r), denoted A ><ra r,
is the completion of Cc(r, A) with respect to the norm
llxllu =sup 117r(x)JI,
where the supremum is over all (cyclic) *-homomorphisms 7r : Cc (r, A) -+
JB(H).
Though it isn't completely obvious, we will soon see that there are lots
of representations Cc(r, A) -+ JB(H). (In particular, II · llu really is a norm,
as opposed to seminorm, on Cc(r, A) and hence we have a natural inclusion
Cc(r, A) c A ><ra r.) Evidently our definition implies the following universal
property.
Proposition 4.1.3 (Universal property). For every covariant representa-
tion ( u, 7r, H) of a I'-C -algebra A, there is a -homomorphism (7: A ><ra r -+
JB(H) such that
O'(Lass) = LK(as)U 8 ,
sEr sEr
for all l:sEr a 8 s E Cc(r, A).
To define the reduced crossed product, we begin with a faithful repre-
sentation Ac JB(H). Define a new representation of A on'}-{ @£^2 (r) by
K(a)(v@ 8g) = (ag-1(a)(v))@ 8g,where {8g}gEG is the canonical orthonormal basis of £^2 (r). Under the iden-
tification'}-{@ £^2 (r) ~ E9gEr '}-{ we have simply taken the direct sum repre-
sentation
K(a) = E9a;^1 (a) E JB(E9H).
gEI' gEI'
The point of doing this is that now the left regular representation of r
spatially implements the action a: for all elementary tensors we have
(1 @A 8 )7r(a)(l@ A;)(v@ 8g) = (1@ A 8 )7r(a)(v @8s-1g)
= (1@ A 8 )((ag-1 8 (a)(v))@ 08 -ig)
= (a~-1 8 (a)(v))@ 8g
= (ag-1(a 8 (a))(v))@ 8g
= 7r(a 8 (a))(v@8g)·