220 6. Amenable Traces
(3) the positive linear functional μ 7 : A 0 A^0 P --+ <C is continuous with
respect to the minimal tensor product norm;
(4) the product morphism w 7 x w~P: A 0 A^0 P--+ IIB(L^2 (A, T)) is contin-
uous with respect to the minimal tensor product norm;
(5) for any faithful representation AC IIB(H) there exists a u.c.p. map
qi: IIB(H)--+ w 7 (A)" such that q?(a) = w 7 (a) for all a E A.^7Proof. (1) ==? (2): Let Ac IIB(H) be a faithful representation. Since Tis an
amenable trace, we can find a state 1jJ on IIB(H) which extends T and such that
'ljJ(uTu) = 1/J(T) for all unitaries u EA and operators TE IIB(H). Since the
normal states are weak- dense in the set of all states on IIB(H), we can find
a net of positive operators h>. E Si such that Tr(h>.T) --+ 1/J(T) for all T E
IIB(H). Since 1/J(uTu) = 1/J(T), it follows that Tr(h>.T) - Tr((uh>,u)T) --+ 0
for every TE IIB(H) and unitary u E A. In other words, for a fixed unitary
u E A, h>. - uh>.u --+ 0 in the weak topology on S1. Hence, by the Hahn-
Banach Theorem, some convex combinations of the elements {h>. - uh>.u}
will tend to zero in the L^1 -norm.
As we saw in the proof of Lemma 2.3.4, a simple direct sum trick implies
that we can extend this to a finite set of unitaries in A.^8 That is, if i c A
is a finite set of unitaries, then for every E > 0 we can find a positive
trace class operator h E S1 such that Tr(h) = 1, I Tr(uh) - T(u)J < E and
llh-uhu* 111 < E for all u E i. Since finite-rank operators are norm dense in
S1, we may further assume that his finite-rank with rational eigenvalues.
Applying Lemma 6.2.5 to bigger and bigger finite sets of unitaries and
smaller and smaller e's, one constructs a net of u.c.p. maps cpn: IIB(H) --+
Mk(n)(<C) such that tr(cpn(u))--+ T(u) and I tr(cpn(uu*) - cpn(u)cpn(u*))I--+ 0
for every unitary element u EA. From Lemma 6.2.6 it follows that
llcpn(au) - CfJn(a)cpn(u)ll2,tr--+ 0
for every unitary element u E A and we are finished.
(2) ==? (3): Since we must show that for every x EA 0 A^0 P,
lμT(x)I '.S llxllmin,
it will suffice to exhibit a net of states which converge to μ 7 pointwise and
are defined on all of A ® A^0 P.
Let cpn: A--+ Mk(n)(<C) be a net of u.c.p. maps as in condition (2).
Note that we can also regard these maps as sending A^0 P to Mk(n) (<C)^0 P(^7) This is also equivalent to the existence of a single representation with this property.·
(^8) More precisely, one considers then-fold direct sum of Si and, for a fixed n-tuple of unitaries
u1, ... ,un EA, one shows that (uih>.ui - h>,, ... ,·unh>,u~ - h>.) is converging weakly to zero.
Hence a convex combination of the h>. 'swill simultaneously bring all the norms ffuih>.u: - h>. [[1,
i = 1, ... , n, close to zero.