# Markovian processes on mutually commuting von Neumann algebras

Banach Center Publications (1998)

- Volume: 43, Issue: 1, page 111-118
- ISSN: 0137-6934

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topCecchini, Carlo. "Markovian processes on mutually commuting von Neumann algebras." Banach Center Publications 43.1 (1998): 111-118. <http://eudml.org/doc/208830>.

@article{Cecchini1998,

abstract = {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.},

author = {Cecchini, Carlo},

journal = {Banach Center Publications},

keywords = {Markov state; commuting von Neumann algebras; Tomita theory; generalized conditional mean values; noncommutative Markov chain; modular automorphisms; Markov triple},

language = {eng},

number = {1},

pages = {111-118},

title = {Markovian processes on mutually commuting von Neumann algebras},

url = {http://eudml.org/doc/208830},

volume = {43},

year = {1998},

}

TY - JOUR

AU - Cecchini, Carlo

TI - Markovian processes on mutually commuting von Neumann algebras

JO - Banach Center Publications

PY - 1998

VL - 43

IS - 1

SP - 111

EP - 118

AB - 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.

LA - eng

KW - Markov state; commuting von Neumann algebras; Tomita theory; generalized conditional mean values; noncommutative Markov chain; modular automorphisms; Markov triple

UR - http://eudml.org/doc/208830

ER -

## References

top- [1] L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245-273. Zbl0483.46043
- [2] C. Cecchini, On the structure of quantum Markov processes, Quantum Probability and Related Topics Vol. IX, 149-157, World Scientific.
- [3] C. Cecchini, Stochastic coupling for von Neumann algebras, preprint. Zbl0970.46047

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