On Ergodic Type Theorems for Strictly Weak Mixing C∗-dynamical Systems

It is known (see [16]) that there are several notions of mixing (i.e. weak mixing, mixing, completely mixing e.c.t.) of measure preserving transformation on a measure space in the ergodic theory. It is important to know how these notions are related with each other. A lot of papers are devoted to this topic. In [1] a rather thorough study of the mentioned concepts of mixing for Markov operators were given. These definitions depend only on the equivalence class of the measure m and on P given on L(X,μ). In the classical ergodic theory of probability-measure preserving transformations it is well known that a transformation is weakly mixing if and only if it has continuous spectrum, or, alternatively, if and only if its Cartesian square is ergodic. The main results of paper [1] concern the corresponding results for Markov operators. There is another recent paper [4], where some relations between the notions of weak mixing and weak wandering have been studied. In this paper we deal with noncommutative analog of the mentioned notions mixing for quantum dynamical systems over C-algebras. Here by quantum dynamical systems we mean a linear, positive mapping T of C-algebra A, with a state φ, into itself. It is known (see [7], sec.4.3, [25],[22]) that the theory of quantum dynamical systems provides convenient mathematical description of the irreversible dynamics of an open quantum system. This motivates an interest to the study of conditions for a dynamical system to induce approach to a stationary state, of reflect subjects such as irreducibility (i.e. ergodicity, mixing) and ergodic theorems (see for example, [2],[11],[14]). A lot of papers (see, [10], [12],[18],[19],[26]) were devoted to the investigations of mixing properties of dynamical systems. Very recently in [21] certain relations between ergodicity, weak mixing and uniformly weak mixing conditions of C-dynamical systems have been investigated. It is known [26],[17] that strict ergodicity (or uniform ergodicity) of a dynamical system is stronger than ergodicity. Therefore, it is natural to ask, how this notion is related with mixing