Noncommutative Mathematics for Quantum Systems

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