Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 155

ψn(ι(a)ul) =

n

∑

k= 1

1 ≤k−l≤n

ek−l,k⊗αl−k(a). (2.4.2)

In particularψn(ι(a)ul) ∈ Mn(A)and Remark 2.4.1 implies that
ψn:AoαZ→Mn(A). The mapφn:Mn(A)→AoαZis defined
by the following explicit formula:

φn(er,s⊗a)=

1

n

u−rι(a)us=

1

n

ι(αr(a))us−r,r,s=1,... ,n,a∈A.

(2.4.3)

Unitality ofψnis easy to check. Complete positivity follows from
the fact thatψn can be written as a composition of completely

positive mapsι(n) andx 7→ V∗xV, where x ∈ Mn⊗B(^2 (Z) ⊗H)≈B(Cn⊗^2 (Z)⊗H)andV∈B(^2 (Z)⊗H;Cn⊗^2 (Z)⊗H),
withV = [u^1 ,···,un]. The defining formulas (2.4.2) and (2.4.3)

and the definition ofβ ̃allow us to check easily that we have

ψn◦ ̃β=β(n)◦ψn, β ̃◦φn=φn◦β(n). (2.4.4)

Further fora∈Aandl∈Zwe have

φn

(

ψn(ι(a)ul)

)

=φn







n

∑

k= 1

1 ≤k−l≤n

ek−l,k⊗αl−k(a)







=

n

∑

k= 1

1 ≤k−l≤n

φn

(

ek−l,k⊗αl−k(a)

)

=

1

n

n

∑

k= 1

1 ≤k−l≤n

ι(αk−l(αl−k(a)))ul

=

1

n

n

∑

k= 1

1 ≤k−l≤n

ι(a)ul=

n−|l|

n

ι(a)ul.

This fact together with Remark 2.4.1 implies (remember that unital
completely positive maps are automatically contractive) that for
eachx∈AoαZ

lim

n→∞

‖φn(ψn(x))−x‖=0.