148 Noncommutative Mathematics for Quantum Systems
The algebraA(X)was first introduced and studied by Powers. It
can be identified with a twisted version of theC∗-algebra of the
compact group
⊕
ZZ^2. In particular it is nuclear, and possesses a
faithful tracial stateτ, uniquely determined by the formula
τ(si) ={
1 if i=0,
0 if i∈Z\{ 0 }.In general the properties of the quantum dynamical system
(A(X),σX)depend on the choice ofX, but one can show that for
almost every (with respect to the Bernoulli measure on
2 N ≈ ∏n∈N{0, 1}) nonperiodicX ⊂Nthe following properties
hold:
(i)τis the uniqueσX-invariant state onA(X);
(ii) the CNT entropy ofσXwith respect to the traceτvanishes:
hτ(σX) =0;
(iii) the CNT entropy ofσX⊗σXwith respect to the traceτ⊗τ
does not vanish:hτ⊗τ(σX⊗σX) =log 2.Note that in particular the binary shifts provide counterexamples
for the additivity of the CNT entropy:hτ⊗τ(σX⊗σX) 6 =hτ(σX) +
hτ(σX). For the proof of these properties we refer to Chapter 12 of
[NS].
Using these properties and some careful, but not too difficult von
Neumann algebra type arguments (see Section 2.5) one can prove
the following result.
Theorem 2.3.7 ([Sk 1 ]) For almost every nonperiodic setX ⊂N
the binary shiftσXsatisfies htc(σX) = 0 <log 2 2 ≤ht(σX).
Sketch The upper estimate follows immediately from property
(iii) above, subadditivity of Voiculescu’s topological entropy with
respect to tensor products and the fact that it dominates the CNT
entropy.
For the lower estimate we assume thatC be a commutative
σ-invariantC∗-subalgebra ofA, writeσX|C=σCand note that due
to the classical variational principle (which says that the
topological entropy for a classical dynamical system is equal to the
supremum of the corresponding measure entropies with respect to
invariant measures for the system, see Chapter 8 of [Wa]) it
suffices to show that the classical measure entropyhφ(σC) =0 for
everyσC-invariant stateφonC(which is in fact a measure on the