Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 145

This implies that rcp(Ω(ln),e)≤ClNm,

log rcp(Ω(ln),e)≤Cl+mlogN=Cl+ ((k− 1 )n+l)logN

and further

lim sup

n→∞

(

1

n

log rcp(Ω(ln),e)

)

≤(k− 1 )logN.

Kolmogorov–Sinai property of the Voiculescu entropy (Proposition
2.2.11) ends the proof.

The above result is a simplified version of Theorem 2.2 from [SZ 2 ].
The proof is partially inspired by the article [BG].

2.3.2 The Voiculescu entropy of the noncommutative shift and
related questions

Theorem 2.3.1 allows us in particular to obtain the following result,
originally proved, via different methods, in [Ch].

Theorem 2.3.4 ([Ch]) LetN∈N,N≥2 and letΦ:ON→ONbe
the shift endomorphism. Then htΦ=logN.

Proof Theorem 2.3.1 and Propositions 2.2.3 imply that

htΦ≤logN.

On the other hand Theorem 2.2.4 and the monotonicity of the
Voiculescu entropy allow us to deduce that

htΦ≥htop(T),

whereTdenotes the shift onCN. Ashtop(T) =logN(Proposition
2.2.7), the proof is finished.

The proof of the above theorem is a standard example of
computing the Voiculescu entropy via ‘classical subsystems’. Even
for relatively simple permutative endomorphisms of Cuntz
algebras this method may need to be fine-tuned, as the following
exercise shows (we will later see in Theorem 2.3.7 that in other
cases this approach may simply fail).

Exercise 2.3.2 LetN=2. Consider the endomorphismρ:O 2 →
O 2 given by the prescription

ρ(S 1 ) =S 1 S 2 S∗ 1 +S 1 S 1 S∗ 2 , ρ(S 2 ) =S 2.