Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras

On quantum quadratic operators of M2(C) and their dynamics

“…However, such kind of operators does not cover the case of quantum systems. Therefore, in [5,6] quantum quadratic operators acting on a von Neumann algebra were defined and studied. Certain ergodic properties of such operators were studied in [21,22].…”

On infinite dimensional quadratic Volterra operators

“…It includes as a particular case of quadratic stochastic operators. In [5,11] some ergodic and stability properties of such operators were studied. But it would be more interesting to investigate one of the simplest case in which that operators act on infinite dimensional algebras.…”

“…But these operators do not occupy quantum systems, so it is natural to investigate quantum quadratic operators. In [4,5] a notion of quantum quadratic stochastic operators defined on von Neumann algebra has been introduced. It includes as a particular case of quadratic stochastic operators.…”

On Marginal Markov Processes of Quantum Quadratic Stochastic Processes

“…were defined on a von Neumann algebra and studied certain ergodic properties ones. In [3] it is obtained necessary and sufficient conditions for the validity of the ergodic principle for q.q.s.p. From the physical point of view this means that for sufficiently large values of time the system described by such a process does not depend on the initial state of the system.…”

“…So, it is naturally to define a concept of quantum quadratic processes. In [3,10] quantum (noncommutative) quadratic stochastic processes (q.q.s.p.) were defined on a von Neumann algebra and studied certain ergodic properties ones.…”