4.8. Maps on reduced amalgamated free products 163and such that V;an(a)Vn = PK7r(a)PK, Vn7r(,6n(x))V; = Vn V;xVn v; for all
a EA and x E Mk(n)(C).Proof. Let n be the set of all (possibly noncontractive) c.p. maps () on A
of the form()= ,6 o a, where a is such that
a( a) = PK17r( a )PK' EB a' (a) E lffi(JC' EB £)
for some finite-dimensional Hilbert spaces JC',[, with JC c JC' c 1-{ and some
u.c.p. map a': A ---t JIB(£) and ,6: lffi(JC' EB£) ---t A is such that
7r(,6(l))PK =PK and PK7r(,6(x))PK = PKXPK
for all x E lffi(JC' EB£). (We apologize for the abuse of notation.) Observe that
n is a convex set. Indeed, if ()i = ,6i o ai ( i = 1, 2) with ai: A ---t lffi(JC~ EB £,i)
and 0 < t < 1 are given, then t()1 + (1 - t)()2 = ,6 o a, where
a(a) = PK17r(a)PK' EB (a~(a) EB a;(a)) E lffi(JC' EB (£1 EB £2))
for the finite-dimensional Hilbert space JC' spanned by JC]_ and JC; and
,6(x) = t,61(PKi63.Ci x Pq 63.CJ + (1 - t),62(PK~$.C 2 x PK~ 63 .c 2 ).
We claim that it suffices to show the identity map idA of A is in the
point-weak closure of n. Indeed, if this were true, then, taking convex
combinations, we could find a net ()n = ,6n o an in D which converges to idA
in the point-norm topology (Lemma 2.3.4). In particular, bn = ()n(l) ~ 1 for
sufficiently large n. Hence, we can approximate (and replace) ,6n with the
u.c.p. map fin(·)= b;;,^112 ,6n( · )b;;,^112. Since bn = ,6n(l) is such that 7r(bn) =
PK+ Pf7r(bn)P:f, the u.c.p. map fin still satisfies PK7r(fin(x))PK = PKXPK
for all x, showing the fin's to be as desired.
It is left to show that idA is in the point-weak closure of D. Let p EA
be the central projection supporting the irreducible representation ( 7r, 11).
It follows that pA = lffi(H) canonically. Since A is nuclear, there exist nets
of u.c.p. maps a~: A ---t Iffi(Cn), ,6~: IIB(Cn) ---t (1-p)A*, with Cn finite-
dimensional, such that the net ,6~ o a~ converges to the -homomorphism
A ---t ( 1 - p) A, a 1--+ ( 1 - p) a, in the point-norm topology. Let JCn be an
increasing net of finite-dimensional subspaces of 11 containing JC and with
dense union (all index sets may be assumed the same). We define c.p. maps
an: A ---t Iffi(JCn EB Cn) and ,6n: lffi(JCn EB Cn) ---t A by
an(a) = PKn7r(a)PKn EB a~(a) E lffi(JCn EB Cn)
and
,6n(x) = PKnXPKn EB ,6~(P.cnxP.cn) E lffi(H) EB (1-p)A =A.
It is clear that the net ,6n o an converges to idA in the point-ultraweak topol-
ogy and that 7r(,6n(x)) = PKnXPKn for all x. By the bijective correspondence