Quantum ergodicity of \(C^*\) dynamical systems (Q1918108)

The author defines a notion of quantum, or non-commutative, ergodicity for a class of \(C^*\)-dynamical systems \(({\mathcal A}, G, \alpha)\) called quantized GNS systems. Such a system possesses a classical limit state \(\omega\), which induces a classical limit system by the GNS construction. The criterion for quantum ergodicity is that the time average \(\langle A\rangle\) of an observable \(A\in {\mathcal A}\) equals the ``space average'' \(\omega(A)I\) plus an error \(K\) which is negligible in the classical limit. Finally, (i) it is proved that ergodicity of \(\omega\) is a sufficient condition for quantum ergodicity of \(({\mathcal A}, G, \alpha)\) if the classical limit system is Abelian, (ii) a conditional converse is given, and (iii) a number of applications are discussed.