Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 137
Aψid //AMnφ>>Further forΩ ⊂⊂Aande>0 we will write(φ,ψ,Mn) ∈CPA
(A,Ω,e)if(φ,ψ,Mn)∈CPA(A)and
∀a∈Ω‖φ◦ψ(a)−a‖<e.The last condition means that the above diagram is ‘commutative
onΩup toe’.
Definition 2.2.9 The algebra A is said to be nuclear, if
CPA(A,Ω,e) 6 =∅for anyΩ⊂⊂A,e>0. In such a case define
rcp(Ω,e):=min{n∈N: (φ,ψ,Mn)∈CPA(A,Ω,ε)}.CommutativeC∗-algebras are nuclear – we will show this fact
below. So are Cuntz algebras and more generally graph
C∗-algebras (this can be shown in several ways – one approach is
based on crossed-product-type constructions similar to those to be
introduced in Section 2.4, another one on exploiting the properties
of the algebraFΛand the fact that it is a fixed point algebra for a
natural action of the circle T on C∗(Λ)). The reduced group
C∗-algebra of a discrete group Γis nuclear if and only if Γ is
amenable(see Section 2.4). More information onC∗-approximations
can be found in the excellent book [BrO]. We will later need the
following simple observation.
Exercise 2.2.6 LetA,Bbe unitalC∗-algebras, letγ:A→Bbe a
ucp map and letΩ⊂⊂A. Show that if eitherAorBis nuclear, then
for eache>0 there existsn∈Nand ucp mapsψ:A→Mnand
φ:Mn→Bsuch that for alla∈Ωthere is
‖γ(a)−(φ◦ψ)(a)‖<e.2.2.5 Noncommutative topological entropy (Voiculescu
topological entropy)
Definition 2.2.10 ([Vo]) Let (A,α) be a quantum dynamical
system and assume thatAis nuclear. Noncommutative topological
entropy ofαis the number