1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.1. Crossed products

L a;^1 (a) 0 ep,s-lp EA 0 MF(<C).

pEFnsF

Now if x =I: asAs E Cc(r, A) c IIB(?i 0 .e^2 (r)), then we have

(10 P)x(l 0 P) = L L a;^1 (as) 0 ep,s-lp EA 0 MF(<C)

sErpEFnsF

119

and thus the norm of (10 P)x(l 0 P) does not depend on the embedding

Ac IIB(?i). D

The following description of positive elements is sometimes handy.

Corollary 4.1.6. An element x = L:sa asAs E Cc(r, A) is positive in

A ><la,r r if and only if for any finite sequence s1, ... 'Sn E r, the operator

matrix [a;/(asis-:-1)]i,j E Mn(A) is positive.

J

Proof. Since an operator is positive if and only if its compression by any

projection is positive, the result follows from a calculation above:

(10 P)x(l 0 P) = L L a;^1 (as) 0 ep,s-lp EA 0 MF(<C).

sErpEFnsF

Indeed, if F = {s1, ... , sn}, then we can identify this double sum with
the operator matrix in the statement of the corollary. (Let p = Si and
Sj = s-^1 p.) D

Here is a C*-dynamical version of Fell's absorption principle, with iden-

tical proof.

Proposition 4.1.7 (Fell's absorbtion principle II). If (u, idA, 1-t) is a covari-

ant representation (i.e., Ac IB\(1-t) and the action a is spatially implemented

in this representation), then the covariant representation

(u 0 >., idA 01, 1-t 0 .e^2 (r))

is unitarily equivalent to a regular representation. In particular, we have a

natural *-isomorphism

C* ( ( u 0 >.) (r), A 0 1) ~ A ><1 a,r r.

Proof. Let ( u, idA, 1-t) be a covariant representation and define a unitary U
on 1-t 0 .e^2 (r) by U(e 0 8t) = ute 0 8t. One checks that

U(l 0 As)U =(us 0 As) and U(L at^1 (a) 0 et,t)U =a 01
t
for every s E r and a E A. D

We close this section with the existence of conditional expectations.

First, a lemma.