Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 165

Fix now a normal functionalγ∈M∗⊂M∗and define for each
n∈N,x∈M

sn(γ,x) =

1

n

n− 1

∑

k= 0

γ◦αk(x).

Assume that there existy,z∈Msuch thatγ(x) =〈πω(y)Ω,πω(x)
πω(z)Ω〉forx∈M. Then we have for eachk∈N 0

γ(αk(x))=〈πω(y)Ω,πω(αk(x))πω(z)Ω〉=ω(y∗αk(x)z)=ω(zy∗αk(x))

=〈πω(yz∗)Ω,πω(αk(x))Ω〉=〈πω(yz∗)Ω,Uk(πω(x)Ω)〉,

where we used the fact thatωis tracial. Thus for this particularγ
using the equality (2.6.2) we obtain that

sn(γ,x)=

〈

πω(yz∗),

1

n

n− 1

∑

k= 0

Uk(πω(x)Ω)

〉

n→∞

−→〈πω(yz∗),P(πω(x)Ω)〉.

As it is easy to see that‖sn(γ,x)‖ ≤ ‖γ‖‖x‖for anyγ∈M∗and
x∈M, linearity of the mapγ7→sn(γ,x), density ofπω(M)ΩinHω
and the last statement of Proposition 2.5.9 implies that the sequence
of complex numbers(sn(γ,x))∞n= 1 is in fact convergent for anyγ∈
M∗,x∈M. Denote the relevant limit by ̃s(γ,x), fix for the moment
x∈Mand consider the map

M∗ 3 γ7→ ̃s(γ,x).

It is easy to check using again the properties listed in the last
paragraph that this map is a bounded linear functional onM∗(of
norm not greater than‖x‖). AsM= (M∗)∗there existsmx ∈M
uniquely determined by the equality

γ(mx) = ̃s(γ,x), γ∈M∗. (2.6.3)

PutΦ(x) =mx, x∈Mand note that formulas (2.6.1) and (2.6.3)
coincide. The fact that so-defined mapΦis linear and contractive is
easy to verify.
To show thatΦ(M)⊂Fixαit suffices to note that for everyx∈M
andγ∈ M∗we haveγ(Φ(x)) = γ(α(Φ(x)), or in other words
that ̃s(γ,x) = ̃s(γ◦α,x). The last formula follows easily once we
write down explicit limits. On the other hand, ifx ∈ Fixαthen