140 Noncommutative Mathematics for Quantum Systems
Proposition 2.2.11 (Kolmogorov–Sinai property, [Vo]) LetAbe a
nuclearC∗-algebra,α:A→Aa unital∗-homomorphism. If(Ωi)i∈I
is a family of finite subsets ofAsuch that
⋃
i∈I,n∈NΩ
(n)
i is linearly
dense inA, then
htα= sup
e>0,i∈I
lim sup
n→∞
(
1
n
log rcp(Ω(in),e)
)
.
Note that in particular we can apply the above result in the
situation where
⋃
i∈IΩiis linearly dense inA.
Finally, we introduce the notion of noncommutative topological
pressure owing to Neshveyev and Størmer. LetAbe a nuclear
C∗-algebra,α :A → Aa unital∗-homomorphism anda ∈Abe
self-adjoint. Define for eachn∈Nthe elementa(n)=∑in=− 01 αi(a)
and thenoncommutative partition function(e>0,n∈N)
Zα,n(a,Ω,e) =inf{Tr eψ(a
(n))
:(φ,ψ,Mk)∈CPA(A,Ω(n),e)},
where Tr denotes the canonical (not normalized) trace onMk(that
is, Tr(q) =1 for any minimal projectionq∈Mk), and for a matrix
M∈Mkwe write eM=∑∞n= 0 M
n
n!. Define
Pα(a,Ω,ε) =lim sup
n→∞
1
n
log(Zα,n(a,Ω,ε)),
and thenoncommutative pressure
Pα(a) = sup
ε>0,Ω⊂⊂A
Pα(a,Ω,ε).
Again, as in the classical case, for any quantum dynamical
system(A,α)we have
htα=Pα( 0 ).
Further, for any classical dynamical system (X,T) and a
real-valued f ∈ C(X) we have P(T,f) = PαT(f); the
noncommutative dynamical pressure satisfies also a version of the
Kolmogorov–Sinai property.
Finally, note that both the notions of noncommutative
topological entropy and pressure can be naturally extended to
unital completely positive maps of nuclearC∗-algebras (which we
will use in the following section). More generally, one can study
similar ideas for other mathematical categories, as soon as one