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On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras

F. M. Mukhamedov


Abstract: We give an expansion of a quantum quadratic stochastic process (q.q.s.p.) into a so-called fibrewise Markov process and prove that, conversely, such an expansion uniquely determines the quantum quadratic stochastic process. As an application, we give a criterion (in terms of this expansion) for the q.q.s.p. to satisfy the ergodic principle. Using this result, we prove that a q.q.s.p. satisfies the ergodic principle if and only if the associated Markov process satisfies this principle. The expansion obtained is used to introduce a new notion of conjugacy of two q.q.s.p.'s and to study the relation between this notion and the ergodic principle.

UDC: 517.98+519.21

MSC: 46L53, 47D07, 60G07, 81S25

Received: 13.02.2003

DOI: 10.4213/im506

 English version: Izvestiya: Mathematics, 2004, 68:5, 1009–1024

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