158 Noncommutative Mathematics for Quantum Systems
are measurable and for each measurable A ⊂ Y there is
μ(T(A)) =μ(A)). In this context we can also speak of entropy (as
mentioned before), but we want to focus on the ergodic-type
results, describing the long-time properties of the system.
The following two results are specific instances of the two
cornerstones of the classical ergodic theory: von Neumann’s mean
ergodic theorem and Birkhoff’s individual ergodic theorem.
Theorem 2.5.1 (von Neumann) Let((Y,μ),T)be a measurable
dynamical system. Then the formula
UT(f) =f◦T, f∈L^2 (Y,μ)defines a unitary operator onL^2 (Y,μ). Write FixUT := {f ∈
L^2 (Y,μ):UT(f) = f}and letP∈B(L^2 (Y,μ))be the orthogonal
projection onto FixUT. Then for eachf∈L^2 (Y,μ)
1
nn− 1
∑
k= 0(UT)k(f)n−→→∞P f (2.5.1)inL^2 -norm.
Theorem 2.5.2 (Birkhoff) Let ((Y,μ),T) be a measurable
dynamical system. Then for each f ∈ L^1 (Y,μ) the sequence
(^1 n∑nk=− 01 f◦Tk)∞n= 1 is almost surely convergent (to a function in
L^1 (Y,μ)).
For a very accessible presentation of the proofs of these classical
results and far-reaching extensions (mostly in the classical context)
we refer to the book [Kre]. Here we just mention that the first
theorem consists of two parts: the first is the construction of the
unitary operatorUT(sometimes called theKoopman representation
of((Y,μ),T)), and the second is a general statement valid for all
Hilbert space unitaries (in fact all Hilbert space contractions).
Later we will discuss certain extensions of the ergodic theorems
involving the simultaneous study of several iterations of a given
mapT.
2.5.2 GNS construction and the passage from topological to
measurable noncommutative dynamical systems
In Sections 2.1–2.4 we investigated certain properties of
topological noncommutative dynamical systems. Now we will