Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 171
Asγwas arbitrary, this implies thatξis supported at most one
point ofΓ, and this point is easily seen to be necessary equal toe(as
ψhas no non-trivial fixed points). Thusξ∈Cδe.
Finally, suppose thatx∈VN(Γ)is a fixed point forαψ. Then
Uψ∗xδe=Uψ∗xUψδe=αψ(x)δe=xδe.Thus,xδeis a fixed vector forUψ∗, so also forUψ. This means that
xδe=μδefor someμ∈C. However,xcommutes with all the right
shifts (see the proof of Proposition 2.5.10), so we can deduce that
xδγ=μδγfor anyγ∈Γ. Thusx=μ1.
Now the fact that we can construct counterexamples to a potential
generalization of Theorem 2.6.6 to noncommutative systems is
based on the following purely group-theoretic result.
Lemma 2.6.9 ([AET]) LetA⊂Z. Then there exists a groupΓwith
elementsγ 0 ,γ 1 ,γ 2 ,γ 3 ∈Γand an automorphismψofΓsuch that
no power ofψhas non-trivial fixed points and ifr∈Zthen
γ 0 Tr(γ 1 )T^2 r(γ 2 )T^2 r(γ 3 ) =eif and only ifr∈A.Proof See Appendix B to [AET].
The group in question is obviously highly noncommutative. Its
construction is based on the concept ofsquare groupsand uses
certain highly non-trivial combinatorial and geometric techniques.
We can now state precisely the result we mentioned in the first
line of this section.
Theorem 2.6.10 ([AET]) There exists a von Neumann algebraM
with a faithful normal tracial stateτand an ergodic automorphism
α ∈ Aut(M,τ) such that (M,τ,α) does not enjoy order 4
convergence in norm.
Exercise 2.6.2 Deduce Theorem 2.6.10 from Lemmas 2.6.8 and
2.6.9.
On the other hand, note that to ensure that(M,ω,α)enjoys order
kconvergence in norm for anyk∈Nit suffices to assume thatωis
a tracial state and the system isasymptotically commutative, that is,
for allx,y∈M