118 4. ConstructionsHence we get an induced covariant representation (10.) X7r, called a regular
representation.^2
Definition 4.1.4. The reduced crossed product of a C* -dynamical system
(A, r, a), denoted A ><1a,r r, is defined to be the norm closure of the image
of a regular representation Cc(r, A)---+ JB(H Q9 f^2 (r)).
For notational simplicity, we will usually forget about the representation
7r and the fact that we had to inflate the left regular representation of r -
i.e., we often (abuse notation slightly and) denote a typical element x .E
Cc(r, A) c A ><la,r r as a finite sum x = 2=sEr as>-s·
Though the following proposition should come as no surprise, the proof
contains some important calculations.
Proposition 4.1.5. The reduced crossed product A ><la,r r does not depend
on the choice of the faithful representation A C JB(H).Proof. The proof boils down to the fact that there is a unique C* -norm on.
Mn(A), just as in the proof of Proposition 3.3.11. For a finite set F c r,
let PE JB(f^2 (r)) be the finite-rank projection onto the span of {5 9 : g E F}.
Rather than compute the norm of x E JB(H Q9 f^2 (r)), we will cut by the
(infinite-rank) projections 1 Q9 P and show that the norm of the compression
is independent of the representation A c JB(H) - taking a limit over finite
sets in r, we conclude the same for x.
Let {ep,q}p,qEF be the canonical matrix units of PJB(f^2 (r))P ~ Mp(C)
and fix some arbitrary elements a E A and s E r. Let 7r: A---+ JB(H Q9 .e^2 (r))
be a regular representation. Our first claim is that
(1 Q9 P)7r(a) = (1 Q9 P)7r(a)(1 Q9 P) = L a;^1 (a) Q9 eq,q·
qEF
This is clear if one thinks of 7r(a) as a diagonal matrix in JB(ffi 9 Er 1-l); in the
tensor product picture we have
7r(a) = L a;^1 (a) Q9 eq,q,
qEI'where convergence is in the strong operator topology.
Thus we see that
(1 Q9 P)7r(a)(1 Q9 A 8 )(1 Q9 P) = ( L a;^1 (a) Q9 eq,q) (1 Q9 P.\ 8 P)
qEF
= ( L a;^1 (a) Q9 eq,q) ( L 1 Q9 ep,s-lp)
qEF pEFnsF(^2) Regular representations are easily seen to be injective on Cc(I', A); hence the universal norm
really is a norm.