4.4. x )<l r -algebras 131
where equality holds ifs= e. In particular, (T, T) = E(g* *a g) = 1. Note
also that (s *a T, s *a T) = 1 and thus
lls *a T-Tll~ = 112 - (T, s *a T) - (s *a T, T)ll,which will be close to zero whenever l(g**a9)(x, s)I is close to one (uniformly
in x). But this happens whenever the positive-type function his uniformly
close to one, so the proof is complete.
(1) =? (3): This follows immediately from Lemma 4.3.7 and Theorem
4.3.4.
(3) =?- (4): Let B be any unital C*-algebra, but regard it as a r-C*-
algebra with the trivial r-action T. Then, C(X) @Bis a (X )<l r)-C*-algebra
and
(C(X) ><la I') 0max B = (C(X) 0max B) ><la1zn-I'
= ( C(X) 0 B) ><la@T,r r
= (C(X) ><la I') 0 B,where the first equality follows from universal considerations, the second
uses nuclearity of C(X) and our assumption that reduced and universal
crossed products are isomorphic, while the last line uses Exercise 4.1.3 and
our assumption one more time. Therefore C(X) ><la r is nuclear.
(4) =? (2): Let a finite subset F c r and c: > 0 be given. Since the
crossed product C(X) ><la,r r is nuclear, there exist u.c.p. maps
cp: C(X) ><la,r I'-+ Mn(C), 'lj;: Mn(C)--+ C(X) ><la,r r and () = 'lj; o cp
such that llB(.As) - Asll < c: for s E F. By construction we have an inclusion
C(X) ><la,rI' C JIB(H@R^2 (r)); hence we may assume that <pis the compression
onto JIB(H 0 @R^2 (F)) for some finite subset F c rand some finite-dimensional
subspace Ho CH (Exercise 3.9.5). It follows that B(.As) = 0 ifs ¢:. _F_F-l
(i.e., if sF n F = 0). Denote by E: C(X) ><la,r r --+ C(X) the canonical
conditional expectation and define h E Cc(X x r) by h(x, s) = hs(x), where
hs = E(B(As)X;^1 ) E C(X). For every s E F, we have
Ill - hsllo(X) = llE((As - B(As)).A;^1 )11:::; II.As - B(As)ll < C:.
Moreover' h is of positive type. Indeed, for any s1' ... ' Sn E r' we have
[a (^8) i :-1(h 8 i (^8) J --:1)]i,j = [a (^8) i :-1(E(B(\i (^8) J --:1)\j 8 :-1))] i i,J ..
= [E ( .A;i B( Asi .A;j )Asj )] ..
i,J
= E( diag(Asu ... , Asn)*B([AsiA;j]i,j) diag(Asu ... , Asn))
since Eis r-equivariant. But this latter matrix is positive since E and() are
c.p. 0