Noncommutative Mathematics for Quantum Systems

164 Noncommutative Mathematics for Quantum Systems

Theorem 2.5.1. Note that the above statements can be viewed as
another way of phrasing of the GNS construction.
There is another possible approach to generalizing the mean
ergodic theorem to the context of von Neumann algebras, which
can be traced back to the paper [KS]. We describe it here briefly in
a simplified version to illustrate the notions and techniques
introduced in Section 2.5.

Theorem 2.6.1 LetMbe a von Neumann algebra, letω∈M∗be a
faithful normal tracial state and letα∈Aut(M,ω). Define Fixα=
{m∈M:α(x) = x}. Then there exists a contractive idempotent
mapΦfromMonto Fixα, which has the following properties:

(i)Φis positive;

(ii)ω◦Φ=ω;

(iii) for allγ∈M∗andx∈M

nlim→∞

1

n

n− 1

∑

k= 0

γ(αk(x)) =γ(Φ(x)). (2.6.1)

Proof Let(πω,Hω,Ω) be the GNS construction for(M,ω) and
define for eachx∈M

U(πω(x)Ω) =πω(α(x))Ω.

ThenUis an isometry on the spaceπω(M)Ω⊂Hω: forx,y∈M

〈U(πω(x)Ω),U(πω(y)Ω)〉=〈πω(α(x))Ω,πω(α(y))Ω〉

=〈Ω,πω(α(x)∗α(y))Ω〉=ω(α(x∗y))

=ω(x∗y) =〈πω(x)Ω,πω(y)Ω〉.

The fact thatπω(M)Ωis dense inHωimplies thatUextends to an
isometry on the whole Hilbert space; using the aforementioned
density and invertibility ofαwe conclude thatUis in fact unitary.
Thus, (the abstract version of) the von Neumann’s mean ergodic
theorem implies that for everyξ∈Hω

1

n

n− 1

∑

k= 0

Uk(ξ)n−→→∞Pξ, (2.6.2)

wherePis a projection inB(Hω).