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.