Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 169
Theorem 2.6.6 Let((Y,μ),T)be a measurable dynamical system,
letM=L∞(Y,μ), letω∈S(M)be induced by the measureμand
letα∈Aut(M,ω)be induced byT. Then for eachk∈Nthe triple
(M,ω,α)enjoys orderkconvergence in norm.
The origins of the above result can be traced to the fundamental
work of Furstenberg. An accessible survey of this and other
classical multi recurrence results, together with the description of
fascinating connections to combinatorics and number theory can
be found in [Kra].
2.6.5 Noncommutative extensions and counterexamples due to
Austin, Eisner, and Tao
A recent paper [AET] contains, apart from several positive results, a
construction of an example which shows that Theorem 2.6.6 cannot
be fully generalized to noncommutative systems.
The algebraMconsidered in the required example is the group
von Neumann algebra of a certain groupΓdefined by generators
and relations; the trace τ is the canonical trace and the
automorphismαis induced from an automorphism ofΓ(via the
construction described in Lemma 2.6.8 below). We first introduce
another definition.
Definition 2.6.7 An automorphismαof a von Neumann algebra
Mis said to beergodicif its fixed point space Fixαconsists only of
the scalar multiples of the unit.
Lemma 2.6.8 LetΓbe a discrete group and letψ:Γ→ Γbe an
automorphism. Then there exists a (unique) normal automorphism
αψ: VN(Γ)→VN(Γ)such that
αψ(λγ) =λψ(γ), γ∈Γ. (2.6.4)The automorphismαψpreserves the canonical traceτ. If, moreover,
for eachk∈Z{ 0 }andγ∈Γ{e}we haveψk(γ) 6 =γ(that is, no
power ofψhas non-trivial fixed points),αψis ergodic.
Proof LetΓandψbe as above. Asψis bijective, we can define a
unitaryUψ on `^2 (Γ) by the continuous linear extension of the
formula
Uψδγ=δψ− (^1) (γ), γ∈Γ;