Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 147
Exercise 2.3.3 Use the above theorem and information contained
in previous sections to reproduce the main result of [BG].
Specifically, letk∈NandM∈Mk({0, 1})be a matrix such that
the associated finite graphΛMhas no sources. Then the Voiculescu
entropy of the ‘generalized shift’ ΦΛM on the Cuntz–Krieger
algebraC∗(ΛM)is equal to the spectral radius of the matrixM,
that is, the limit limn→∞‖Mn‖
(^1) n
.
2.3.4 Automorphism whose Voiculescu entropy is genuinely
noncommutative
Theorem 2.3.4 and Exercise 2.3.2 may seem to suggest that the
Voiculescu entropy could be always simply equal to the
supremum of classical entropies ofallof its classical subsystems.
Define for a quantum dynamical system(A,α)the following
number:
htc(α) =sup
{
ht(α|C):C−commutativeα-invariant
C∗-subalgebra ofA
}
.
General properties of the Voiculescu entropy imply that we always
have htα≥htc(α). An example of an automorphism of a nuclear
C∗-algebra for which the converse inequality does not hold was
given in [Sk 1 ]. We will now briefly describe the relevant model and
list the properties that are crucial for the required argument.
Fix a setX ⊂N 0. ConsiderA(X), the universalC∗-algebra of
thebit-streamassociated toX:A(X)is generated by the family of
operators{si : i ∈ Z}, which are self-adjoint unitaries (that is,
s^2 i = I,si =s∗i for eachi∈ Z– such operators are often called
symmetries) and satisfy the following commutation relations:
sisj= (− 1 )χX(|i−j|)sjsi, i,j∈Z,
whereχX:N 0 → {0, 1}is the characteristic function ofX. Note
that these relations simply mean that the symmetries in question
either commute, or anti-commute, depending on their relative
position with respect toX. As it is only the relative position (that
is, the distance from i to j) which matters, there exists an
automorphismσXofA(X), called thebinary shift, determined by
the prescription
σX(si) =si+ 1 , i∈Z.