146 Noncommutative Mathematics for Quantum Systems
(i) Show thatρis a permutative endomorphism. Deduce further
that htρ≤log 2.
(ii) Find a continuous mapTon the Cantor setC 2 such thatρ|C 2
corresponds toαT(in the sense used in Theorem 2.2.4).
(iii) Show thathtop(T) =0.
(iv) Try to think how one could show thathtρ=log 2. (2.3.6)Equality (2.3.6) was established in [SZ 2 ].
In view of the discussion above and of Exercise 2.2.2 it is natural
to raise the following open problem.
Problem 2.3.5 Does there exist a unitaryU∈FN⊂ONsuch that
htρU > htρU|FN? The evidence so far shows that the answer is
likely negative.
2.3.3 Generalizations to graphC∗-algebras and to
noncommutative topological pressure
Theorem 2.3.4 has a far-reaching generalization, established in
[SZ 1 ], the special case of which we formulate here in the
framework ofC∗-algebras associated to finite graphs. Recall that if
Λ is a finite graph with no sources then CΛ denotes the
commutative C∗-subalgebra of C∗(Λ) isomorphic (via the map
γΛ) to the algebra of continuous functionsC(PΛ∞)on the infinite
path space ofΛ; the isomorphism may be used to identifyΨΛ|CΛ
with the∗-homomorphism induced by the shift mapTΛonPΛ∞
(see Proposition 2.2.5).
Theorem 2.3.6 LetΛbe a row-finite graph with no sources and let
f∈C(PΛ∞)be real-valued. Then the following equality holds:
PΦΛ(γΛ(f)) =P(TΛ,f).The key arguments needed in the proof of the above theorem are
contained in the proof of Theorem 2.3.1; the non-trivial part is
showing the inequality ‘≤’, as the other one follows from
monotonicity of the noncommutative topological pressure. We
recommend the readers a careful analysis of two special cases: the
case whereΛis the graph with one vertex andNedges, that is,
C∗(Λ) = ON, and the case wheref = 0 (so that we simply ask
about the Voiculescu entropy).