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.