Appendix C
Lifting Theorems
Throughout this appendix, we assume that J is a closed two-sided ideal in
a unital C* -algebra B and 7f: B -+ B / J is the quotient map.Definition C.1. Let Ebe an operator system (or a C*-algebra). A c.c.p.
map cp: E -+ B / J is said to be liftable if there exists a c.c.p. map 'If;: E -+ B
such that 7f o 'If; = cp. It is locally liftable if for every finite-dimensional
operator system F C E, the c.c.p. map 'PIF is liftable.If cp is a liftable u.c.p. map, then we can take a unital lifting 'If;. Indeed,
one just has to choose a unital positive linear functional e on E and replace
'If; with 'If;+ (1-'lj;(l))e.
The following result of Arveson is very important - so is its proof.Lemma C.2. Let J be a closed two-sided ideal in a C* -algebra B and let E
be a separable operator system (or a separable C*-algebra). Then, the set of
c. c. p. maps from E into B / J which are liftable is closed in the point-norm
topology.Proof. Let cp: E -t B / J be a c.c.p. map and let 'If;~: E -+ B be c.c.p. maps
such that the sequence 7fO'lj;~ converges to cp in the point-norm topology. Fix
a dense sequence { Xk h in E. Passing to a subsequence if necessary, we may
assume that 117f o 'lj;~(xk) - cp(xk)ll < 1/2n fork:=:; n. We claim that there
exists a sequence of c.c.p. maps 'I/Jn: E-+ B such that 117fo'l/Jn(xk)-cp(xk) II <
1/2n and 11'1/Jn+i(xk)-'l/Jn(xk)ll < l/2n-l fork:=:; n. Once this is established,
the proof is essentially complete. Indeed, since the sequence { 'l/Jn}n of c.c.p.
maps converges point-norm on a dense subset { xk}k, it converges everywhere
to a c.c.p. map 'If;: E-+ B, which is clearly a lifting of cp.
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