Noncommutative Mathematics for Quantum Systems

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



∩

HγHj 1

η

Ψ−m^1 -

*γ ̃

HH

HH

HH

HH

HH

HH

Hj

ψ











*

φ

LetX∈Ap,q(p,q≤l) and letj∈N 0 ,j≤n−1. Then

‖φ◦ψ(ρj(X))−ρj(X)‖

=‖η◦γ ̃◦(id⊗φ 0 ◦ψ 0 )◦Ψm(ρj(X))−(Ψ−m^1 ◦Ψm)(ρj(X))‖

=‖η◦γ ̃◦(id⊗φ 0 ◦ψ 0 )◦Ψm(ρj(X))−η◦γ ̃◦Ψm(ρj(X))

+ η◦γ ̃◦Ψm(ρj(X))−(Ψ−m^1 ◦Ψm)(ρj(X))‖

≤‖(id⊗φ 0 ◦ψ 0 )◦Ψm(ρj(X))−Ψm(ρj(X))‖+

e

2

.

Lemma 2.3.2 implies thatρj(X) ∈ Fp+j(k− 1 ),q+j(k− 1 ). Assume for

example thatp>q. Then, as‖X‖≤1, it follows from Lemma 2.3.3
that

‖(φ◦ψ)(ρj(X))−ρj(X)‖≤

∥

∥∥

∑

μ∈Jp−q

Tμ⊗

(

(φ 0 ◦ψ 0 )(Sμ)−Sμ

)∥∥

∥+

e

2

≤ ∑

μ∈Jp−q

‖Tμ‖‖(φ 0 ◦ψ 0 )(Sμ)−Sμ‖<Np−q

e

4 Nl

+

e

2

=e

(asSμ∈Ap−q,0∈Ωl). Hence, we have shown that

(φ,ψ,MNm⊗MCl)∈CPA(ON,Ω(ln),e). (2.3.5)