Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 141
defines a natural notion of a ‘finite-dimensional approximation up
toe’ and a method of defining the size of this approximation. For
an application of such an idea to the study of Banach space
contractions we refer to [KL].
Finally, as we already mentioned that proving that ifT is a
continuous transformation of a compact space X then the
Voiculescu entropy htαT is equal to htop(T) is difficult, it is a
natural place to state a related open problem.
Problem 2.2.12 LetX be a compact space and let φ : C(X)
→ C(X) be a unital, positive map (recall it is automatically
completely positive). Then Downarowicz and Frej propose in [DF]
three natural definitions of topological entropy ofφ, very roughly
speaking in terms of open covers or separating subsets involving
bothXand finite families of functions inC(X), and show that they
lead to the same numerical value, which we denote byhDF(φ). It is
easy to see, essentially using the methods presented above, that
htφ≤hDF(φ). Do we always have htφ=hDF(φ)?
2.3 Voiculescu Entropy of the Shift and other
Examples
The main aim of this section is to compute explicitly the
Voiculescu entropy and noncommutative topological pressure in
some examples. We will on one hand present cases where the
method of looking at commutative subalgebras yields the final
answer and on the other explain why this technique may
sometimes fail.
2.3.1 Estimating the Voiculescu entropy of the permutative
endomorphisms of Cuntz algebras
We begin by establishing the following estimate for the Voiculescu
entropy of the permutative endomorphisms ofON.
Theorem 2.3.1 ([SZ 2 ]) FixN∈N, letk∈Nand letσ:Jk→ Jk
be a permutation. Denote the permutative endomorphism ofON
corresponding toσ(via the formula (2.2.1) and Theorem 2.2.1) byρ.
Then
htρ≤(k− 1 )logN. (2.3.1)