Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 139

φ([aij]ki,j= 1 ) =

k

∑

i= 1

aiiφi, [aij]ki,j= 1 ∈Mk.

It is easy to see that the mapsψandφare linear, unital, and positive.
Remark 2.1.6 implies that(φ,ψ,Mk)∈CPA(C(X)).

Let theng ∈ Ω(n) =

⋃n− 1

j= 0 α

j

T(Ω), sayg = α

j
T(f)for certain
j∈{0,... ,n− 1 },f∈Ω. Then for anyx∈X

|(φ◦ψ)(g)(x)−g(x)|=|

k

∑

i= 1

φi(x)g(xi)−g(x)|

=|

k

∑

i= 1

φi(x)(g(xi)−g(x))|≤

k

∑

i= 1

φi(x)|g(xi)−g(x)|

= ∑

i=1,...,ks.t.x∈Ui

φi(x)|g(xi)−g(x)|<e,

where the last inequality follows from the fact thatg(xi)−g(x)
= f(Tj(xi))−f(Tj(x))and ifx∈Ui, thend(Tj(xi),Tj(x))< δ.

Thus, we have shown that(φ,ψ,Mk)∈CPA(C(X),Ω(n),e), so that
rcp(Ω(n),e)≤k=s(n,δ).
The inequality (2.2.3) easily follows.

Exercise 2.2.7 Show that ifXis compact, thenC(X)is nuclear. Do
we need to assume thatXis metrisable?

Exercise 2.2.8 Let(A,α)be a noncommutative dynamical system
withAnuclear and letn∈N. Prove thatMn(A)is nuclear and that
htα(n)=htα.

Providing an elementary proof of the inequality

htαT≥htop(T)

turns out to be surprisingly difficult – still the only available
argument is the original one published in [Vo]. It uses a so-
called CNT entropy (due to Connes–Narnhofer–Thirring), a
noncommutative version of the classical measure entropy. It is also
not clear whether one can replace in Voiculescu’s definition the lim
sup by lim inf.
In the following we will need one more important property of
the Voiculescu entropy. It allows us not to check all finite subsets of
a givenC∗-algebra when computing the entropy.