Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 143
Proof To prove the existence ofTμ(without the norm condition), it
suffices to see what happens forX∈Ai,j. For eachν∈ Ji,κ∈ Jj
we have
Ψl(SνS∗κ) = ∑
β,δ∈Jl
eβ,δ⊗S∗βSνS∗κSδ= ∑
β′∈Jl−i,δ′∈Jl−j
eνβ′,κδ′⊗S∗β′Sδ′
= ∑
μ∈Ji−j
( ∑
β′∈Jl−i
eνβ′,κβ′μ)⊗Sμ.
It remains to show the norm estimate for each matrixTμappearing
in the formula (2.3.3). We have:
‖ ∑
μ∈Ji−j
Tμ⊗Sμ‖^2 =‖ ∑
μ∈Ji−j
(Tμ⊗Sμ)∗ ∑
ν∈Ji−j
(Tν⊗Sν)‖
=‖ ∑
μ,ν∈Ji−j
Tμ∗Tν⊗S∗μSν‖=‖ ∑
μ∈Ji−j
Tμ∗Tμ‖,
so for eachμ∈Ji−j
‖Tμ‖^2 =‖Tμ∗Tμ‖≤‖ ∑
ν∈Ji−j
Tν⊗Sν‖^2 =‖Ψl(X)‖^2. (2.3.4)
AsΨlis an injective∗-homomorphism, we obtain
‖Ψl(X)‖=‖X‖,
which together with the estimate (2.3.4) ends the proof.
Analogous lemmas can be established for the casei=jandi<j
(always keepingi,j ≤ l) – we suggest formulating and proving
them as an exercise.
We are ready for the proof of Theorem 2.3.1.
The proof of Theorem 2.3.1 For eachl∈Nlet
Ωl=
⋃l
p,q= 1
Ap,q.
Fixl∈N,e>0. AsONis nuclear, there exists a triple(φ 0 ,ψ 0 ,MCl)
∈CPA(ON,Ωl, 4 N^1 le). Fix nown∈Nand put
Ω(ln)=
n⋃− 1
j= 0
ρj(Ωl).