2.2. Nonunital technicalities 31The right maps to replace the 1/Jn's are slightly technical to describe but
there is nothing deep about the remainder of the proof. First we note that
since e is a unital map,
II lB - ~n O <Pn(lA) II -+ 0.Hence for all sufficiently large n, ~no <Pn(lA) is a positive invertible element
and some standard functional calculus shows that
To ease notation, we let
and
1
Yn = <fSn(lA)2 E Mk(n)(<C).
The u.c.p. maps 'I/Jn: Mk(n) (<C)-+ B we are after are given by
1/Jn(T) = bn,(/Jn(YnTYn)bn,
for all T E Mk(n) (<C). Since
'I/Jn o lfJn(a) = bn;/Jn(<Pn(a))bn
and II lB - bn II -+ o, it follows that 'I/Jn 0 lfJn -+ e in the point-norm topology.
D
Proposition 2.2.7. If A is a C*-algebra, N is a von Neumann algebra and
we are given a unital weakly nuclear map f): A-+ N, then there exist u.c.p.
maps lfJn: A-+ Mk(n) (<C) and 'I/Jn: Mk(n) (<C) -+ N such that 'I/Jn o lfJn -+ e in
the point-ultraweak topology.
Proof. Let <fSn: A-+ Mk(n)(<C) and ;/Jn: Mk(n)(<C)-+ N be c.c.p. maps
whose compositions converge to e in the point-ultraweak topology and, as
before, let lfJn: A-+ Mk(n) (<C) be u.c.p. maps such that
1 1
<Pn(a) = <Pn(lA)2cpn(a)<fSn(lA)2
for all a EA.
Since ;/Jn(<Pn(l)) ::=; ;/;n(l) ::=; lN and ;/Jn(<Pn(l)) -+ lN in the ultraweak
topology, it follows that
is a sequence of positive operators tending ultraweakly to zero. Let Pn be any
sequence of states on Mk(n)(<C) and define linear maps 'I/Jn: Mk(n)(<C)-+ N
by