Noncommutative Mathematics for Quantum Systems

156 Noncommutative Mathematics for Quantum Systems

We are now ready to begin the main part of the proof.
Monotonicity of the Voiculescu entropy with respect to passing to
invariant subalgebras and an observation that (A,β) and

(ι(A),β ̃|ι(A))are isomorphic noncommutative dynamical systems

implies that htβ ≤ ht ̃β. Thus, it suffices to show the reverse
inequality. LetΩ⊂⊂AoαZand letN∈Nbe such that for each
x∈Ωthere is

‖φN(ψN(x))−x‖≤

e

2

.

LetZ = φN(Ω) ⊂⊂ MN(A). Consider nowk ∈ Nand the set
Z(k) =

⋃k− 1

j= 0 (β

(N))j(Z). Let(γ,ρ,Mn) ∈ CPA(Z(k),e
2 ). Then our
claim is that(φN◦γ,ρ◦ψN,Mn)∈CPA(Ω(k),e 2 ), where of course

Ω(k)=

⋃k− 1

j= 0 (β

(N))j(Ω). Indeed, letj∈ {0,... ,k− 1 }andx∈Ω.

Then

‖(φN◦γ◦ρ◦ψN)( ̃βj(x))−β ̃j(x)‖

≤‖(φN◦γ◦ρ◦ψN)(β ̃j(x))−(φN◦ψN◦β ̃j)(x)‖

+‖(φN◦ψN)(β ̃j(x))−β ̃j(x)‖

≤‖

(

φN◦γ◦ρ◦(β(N))j

)

(z)−

(

φN◦(β(N))j

)

(z)‖

+‖ ̃βj((φN◦ψN)(x)−x)‖,

wherez=ψN(x), so thatv:= (β(N))j(z)∈Z(k). Thus further,

‖(φN◦γ◦ρ◦ψN)( ̃βj(x))−β ̃j(x)‖

≤φN((γ◦ρ)(v)−v‖+‖(φN◦ψN)(x)−x‖≤

e

2

+

e

2

=e.

Finally, we see that

lim sup

k→∞

(

1

k

log rcp(Ω(k),e)

)

≤lim sup

k→∞

(

1

k

log rcp(Z(k),

e

2

)

)

≤htβ(N)=htβ,

where the last equality follows from Exercise 2.2.8. AsΩ⊂⊂Aoα
Zande>0 were arbitrary, the proof is finished.

The key ingredient of the proof above is the ‘approximate
invariance’ of the set{1,... ,N}under shifting bylfor largeN.