Geodesic
On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras
Izvestiya. Mathematics , Tome 68 (2004) no. 5, pp. 1009-1024.

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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. 
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     title = {On expansion of quantum quadratic stochastic processes into fibrewise {Markov} processes defined on von {Neumann} algebras},
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F. M. Mukhamedov. On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras. Izvestiya. Mathematics , Tome 68 (2004) no. 5, pp. 1009-1024. http://geodesic.mathdoc.fr/item/IM2_2004_68_5_a5/

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