Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 167
pointwise (almost surely) to a function f if and only if for each
e>0 there existsZ⊂Ysuch thatμ(Y\Z)<eand(fn|Z)∞n= 1 is
uniformly convergent tof|Z.
The above theorem effectively replaces the almost sure
convergence with the uniform convergence on ‘large enough’
subsets. To pass to the noncommutative setting, we first replace
sets by characteristic functions (that is, orthogonal projections in
the algebraL∞(Y,μ)) and note thatfn|Z= fnχZ. This leads to the
following definition, due to Segal.
Definition 2.6.3 LetMbe a von Neumann algebra with a normal
faithful stateω∈M∗. We say that a sequence(xn)∞n= 1 of elements
inMconverges tox∈Malmost uniformly, if for eache>0 there
exists an orthogonal projectionp∈Msuch that
ω(p⊥)<e and nlim→∞‖(xn−x)p‖=0.Note that in fact we could also consider a so-calledbilateral almost
uniformconvergence, replacing above‖(xn−x)p‖by‖p(xn−x)p‖.
2.6.3 Lance’s noncommutative individual ergodic theorem and
some comments on its proof
We are now ready to present a noncommutative counterpart of
Birkhoff’s individual ergodic theorem, owing to Lance.
Theorem 2.6.4 ([La]) LetMbe a von Neumann algebra, letω∈M∗
be a faithful normal state and letα∈Aut(M,ω). Then for everyx∈
Mthe sequence(^1 n∑kn=− 01 αk(x))∞n= 1 is convergent almost uniformly
to an element ofM.
Sketch of the proof The proof of the above theorem, similarly as the
proof of Birkhoff’s Theorem, consists of two main parts. The first
one roughly corresponds to showing that Theorem 2.6.1, or rather
its non-tracial generalization, holds. The second requires proving a
version of a so-called Maximal Lemma, which in this case looks as
follows: for anye>0 andx∈M+such thatω(x) =ethere exists
y∈Msuch that
‖y‖≤2, ω(y)≤ 4 e(^12)