6.2. Amenable traces 221and they are still u.c.p. (Exercise 6.1.7). For notational convenience, we let
'P~P : A^0 P --+ Mk(n) (C)^0 P be the "opposite" maps. Since u.c.p. maps induce
u.c.p. maps on minimal tensor products, we may consider
'Pn 0 'P~p: A 0 A^0 P--+ Mk(n)(C) 0 Mk(n)(C)^0 P.
We'll now construct states μn on Mk(n) (C) 0 Mk(n) (C)^0 P such that
μno ('Pn 0 'P~P)(x)--+ μ 7 (x)
for all x EA 8 A^0 P.
The product map induced by the left and right regular representations
gives an identification (Exercise 6.1.6)
Mk(n)(C) 0 Mk(n)(C)^0 P ~ IIB(L^2 (Mk(n)(C), tr)).
As we have done for A, define a state μnon Mk(n)(C) 0 Mk(n)(C)^0 P by
T 0 S 1---t (7rtr(T)7r~f (S)i, i) = tr(TS).
It follows that
μn('Pn 0 'P~P(a 0 b)) = tr('Pn(a)'Pn(b)),
for all a E A, b E A^0 P. The Cauchy-Schwarz inequality shows that for all
x E Mk(n)(C), I tr(x)I::::; Jlxll2,tr and hence
I tr('Pn(a)'Pn(b)) - tr('Pn(ab))I--+ 0.
Thus we see that
μn('Pn 0 'P~P(a 0 b))--+ r(ab) = μ 7 (a 0 b).
Since the linear span of elementary tensors is dense in A 0 A op, we are done.
(3) =? (4) follows from uniqueness of GNS representations since the
product map A 8 A^0 P--+ IIB(L^2 (A, r)) has a cyclic vector which implements
μT.
(4) =? (5): Apply The Trick to the inclusion A 0 A^0 P c IIB(H) 0 A^0 P and
use Theorem 6.1.4 to control the range of the extending u.c.p. map.
(5) =? (1) is very similar to the proof of Proposition 6.2.2. For every
TE IIB(H) and unitary u EA we have
((uTu)i, i) = (7r 7 (u)(T)7r 7 (u)i, i) = ((T)i, i),
since u is in the multiplicative domain of , (T) E 7r7(A)" and ( · i, i) is
a trace on 7r 7 (A)". Hence T 1---t ((T)i, i) is the desired extending state on
IIB(H). D
Remark 6.2.8. Of course, if A is a separable C*-algebra, then the net in
part (2) of Theorem 6.2.7 can be replaced with a sequence. Moreover, the
same thing is true if A is a von Neumann algebra with separable predual and
r is normal. Indeed, if h>. is a net as in the proof of (1) =? (2) and 'l/J>.( ·) =