144 Noncommutative Mathematics for Quantum Systems
Letm=n(k− 1 ) +l. Apply now the conclusion of Exercise 2.2.6
to the mapΨ−m^1 :Ψm(ON) → ON. There existd∈Nand unital
completely positive mapsγ:Ψm(ON)→Mdandη:Md → ON
such that for alla∈Ψm(Ω(ln))we have
‖η◦γ(a)−Ψ−m^1 (a)‖<
e
2
.
Letγ ̃:MNm⊗ON→Mdbe a unital completely positive extension
ofγ(see Theorem 2.1.7). Consider the following diagram:
ON Ψm(ON)
MNm⊗ON
Ψm(ON) ON
∩
MNm⊗ON
Md
MNm⊗MCl
Ψm -
@id⊗ψ 0
@
@R
id⊗φ 0