On strictly weakly mixing \(C^*\)-dynamical systems (Q941873)

The question of how strict ergodicity is related to mixing conditions is studied in a noncommutative setting. One considers strictly ergodic and strictly weakly mixing \(C^{\ast }\)- dynamical systems. Different notions of mixing (weak, strictly weak, complete mixing, etc.) and strictly ergodic state-preserving \(C^{\ast }\)- dynamical systems are defined and their properties are considered. It is established that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. Some weighted uniform ergodic theorems with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems are also investigated. Illustrative examples of strictly weakly mixing dynamical systems are given.