Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 163
right shift operators byR(Γ). Prove, using Theorem 2.5.6 and
comments after it, that
VN(Γ)′=R(Γ)(some authors writeL(Γ)instead of VN(Γ)).
2.6 Noncommutative Ergodic Theorem of Lance;
Classical and Quantum Multi Recurrence
In this last section we discuss some noncommutative extensions of
the classical ergodic theorems presented in Section 2.5 and also
sketch both classical and quantum results concerning so-called
non-conventional, multiparameter averages.
2.6.1 Mean ergodic theorem(s) in von Neumann algebras
The convergence appearing in the mean ergodic theorem is the
convergence in the Hilbert space norm, and as such still makes
sense when a noncommutative version is considered. We make it
more specific in what follows: note that if we start from the
framework described in Theorem 2.5.7 then the usualL^2 -norm of
an elementf ∈πω(C(X))′′ =L∞(X,μ)can be expressed by the
formula‖f‖ 2 = (ω(f∗f))
1(^2). Similarly, it is not too difficult to verify
that if(M,ω)is a von Neumann algebra with a faithful normal
state, thenMequipped with a new norm: ‖x‖ 2 := (ω(x∗x))
(^12)
(x∈M) becomes a pre-Hilbert space. You may note that the proof
is based on the crucial inequality valid for all states on
C∗-algebras, the Kadison–Schwarz inequality (which is also a
fundamental tool in the GNS construction): ifAis aC∗-algebra
andω∈S(A)then
|ω(x∗y)|≤(ω(xx∗))
1
(^2) (ω(y∗y))
1
(^2) , x,y∈A.
The respective completion is usually denoted as L^2 (M,ω) and
called thenoncommutative L^2 -space associated with(M,ω). Givenα∈
Aut(M,ω)we can then easily verify that the mapx7→α(x),x∈M
extends to a unitary operatorUα on L^2 (M,ω) (see the proof of
Theorem 2.6.1 below). This unitary can be viewed as the
noncommutative Koopman construction for α; it satisfies a
suitably modified counterpart of the formula (2.5.1) of