170 Noncommutative Mathematics for Quantum Systems
it is easy to verify that
Uψ∗δγ=δψ(γ), γ∈Γ.Further for anyγ,γ′∈Γ
Uψ∗λγUψδγ′=Uψ∗λγδψ− (^1) (γ′)=Uψ∗δγψ− (^1) (γ′)
=δψ(γψ− (^1) (γ′))=δψ(γ)γ′=λψ(γ)δγ′.
It follows from the above that the automorphism ofB(^2 (Γ))given byx 7→ Uψ∗xUψrestricts to a normal automorphism of VN(Γ) (this can be shown either via the topological argument or using the identification of the commutant of VN(Γ) provided in Exercise 2.5.3). Denote the resulting restriction by αψ and note that it satisfies the equality (2.6.4). The fact thatαψpreservesτfollows from the equalitiesUψδe = Uψ∗δe=δe. It remains to establish ergodicity ofαψunder the assumption that no power ofψhas non-trivial fixed points. Begin from the following observation: for anyγ∈Γandξ,η∈^2 (Γ)we have:
lim
n→∞
〈ξ,αψn(λγ)η〉n−→ 〈→∞ ξ,τ(λγ)η〉.
Indeed, as all αnψ(λγ) are unitaries (therefore, in particular
contractions), it suffices to verify it forξ = λγ′,η = λγ′′ for
γ′,γ′′ ∈ Γ. Then however the expression on the left of the
displayed limit relation is equal to 1 ifψn(γ′) =γψn(γ′′)and zero
otherwise; in other words, it is equal to 1 only if
γ=ψn(γ′(γ′′)−^1 ). Ifγ=e, then this holds if and only ifγ′=γ′′.
On the other hand ifγ 6 =e, then for the equality above to hold we
cannot haveγ′=γ′′. To finish the analysis of the left-hand side it
suffices to note that an orbit of any non-zero element ofΓunderψ
cannot contain repetitions (as ifψn(t) =ψm(t)thenψn(t)is a fixed
point forψm−n); thus, ifγ 6 =ethen the limit on the left is always
equal to 0. On the other hand the expression on the right is easily
seen to be equal to 1 ifγ=eandγ′=γ′′and to 0 otherwise.
Suppose now thatξ∈`^2 (Γ)is a fixed point for the unitaryUψ.
Then for anyn∈Nandγ 6 =e
〈ξ,λγξ〉=〈Uψnξ,λγUnψξ〉=〈ξ,αn(λγ)ξ〉n−→ 〈→∞ ξ,τ(λγ)ξ〉=0.