Noncommutative Mathematics for Quantum Systems

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).