The aim of this paper is to study markovianity for states on von Neumann algebras generated by the union of (not necessarily commutative) von Neumann subagebras which commute with each other. This study has been already begun in [2] using several a priori different notions of noncommutative markovianity. In this paper we assume to deal with the particular case of states which define odd stochastic couplings (as developed in [3]) for all couples of von Neumann algebras involved. In this situation these definitions are equivalent, and in this case it is possible to get the full noncommutative generalization of the basic classical Markov theory results. In particular we get a correspondence theorem, and an explicit structure theorem for Markov states.
@article{bwmeta1.element.bwnjournal-article-bcpv43i1p111bwm,
author = {Cecchini, Carlo},
title = {Markovian processes on mutually commuting von Neumann algebras},
journal = {Banach Center Publications},
volume = {43},
year = {1998},
pages = {111-118},
zbl = {0937.46059},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv43i1p111bwm}
}
Cecchini, Carlo. Markovian processes on mutually commuting von Neumann algebras. Banach Center Publications, Tome 43 (1998) pp. 111-118. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv43i1p111bwm/
[000] [1] L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245-273. | Zbl 0483.46043
[001] [2] C. Cecchini, On the structure of quantum Markov processes, Quantum Probability and Related Topics Vol. IX, 149-157, World Scientific.
[002] [3] C. Cecchini, Stochastic coupling for von Neumann algebras, preprint. | Zbl 0970.46047