Noncommutative Mathematics for Quantum Systems

168 Noncommutative Mathematics for Quantum Systems

and

1

n

n− 1

∑

k= 0

αk(x)≤y, n∈N.

The proof of the Maximal Lemma is based on a detailed analysis of
properties of certain affine functions defined on a compact convex
subset of a locally convex linear topological space.

Classical Birkhoff’s theorem is usually formulated forL^1 -functions
(and this is how we stated it in Theorem 2.5.2), and not only for
essentially bounded ones. In fact, Lance’s theorem also admits
generalizations to ‘noncommutativeLp-spaces’, but even stating
them reaches beyond the content of our lectures. Interested readers
should consult the paper [JX] and the bibliography therein.

2.6.4 Classical multirecurrence

Recent years have brought a rapid development in the study of the
so-called multirecurrence or ‘multi-ergodic’ theorems. Below we
present the appropriate definitions in the von Neumann algebraic
language.

Definition 2.6.5 LetMbe a von Neumann with a faithful normal
stateωand letα∈Aut(M,ω). We say that(M,ω,α)enjoys order
kconvergence in norm if for anyx 1 ,... ,xk− 1 ∈ Mthe sequence
(yn)∞n= 1 defined by the formula

yn:=

1

n

n

∑

j= 1

(αj(x 1 ))(α^2 j(x 2 ))···(α(k−^1 )j(xk− 1 )), n∈N,

converges inL^2 (M,ω) – in other words it satisfies the Cauchy
condition in‖·‖ 2 -norm, that is,

n,limm→∞ω((yn−ym)∗(yn−ym)) =0.

For classical measurable dynamical systems (M = L∞(Y,μ),
α = αT) the latter condition means simply that the sequence
(yn)∞n= 1 converges inL^2 (Y,μ). Using Lance’s ergodic theorem one
can show that each triple(M,ω,α)enjoys order-2 convergence in
norm. An important theorem of Host and Kra from [HK] can be
reformulated in the following way.