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Stochastic transitions on preduals of von Neumann algebras

Part of the Lecture Notes in Mathematics book series (LNM,volume 1442)

Abstract

The aim of this note is to prove a result on canonical state extensions from a von Neumann subalgebra to a von Neumann algebra in which it is contained. In its light the notion of a stochastic coupling for von Neumann algebras as introduced by the first named author in [5] (cf. also [1]) and the notion of transition operators introduced by the second named author in [7] appear to be particular (indeed extreme) cases of a more general theory.

Keywords

  • Hilbert Space
  • Closed Subspace
  • Faithful Representation
  • Quantum Probability
  • Weak Operator

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Bibliography

  1. L. Accardi: Cecchini’s transition expectations and Markov chains. Preprint (to be published in the proceedings of the Rome year on Quantum Probability 1986/1987).

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  2. C. Cecchini, D. Petz: State extensions and a Radon-Nikodym theorem for conditional expectations on von Neumann algebras. Preprint (to be published in Pac. J. Math.).

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  3. C. Cecchini, D. Petz: Classes of conditional expectations over von Neumann algebras. Preprint (to be published in J. Funct. Anal.).

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  4. C. Cecchini: An abstract characterization of ω-conditional expectations. Preprint.

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  5. C. Cecchini: Stochastic couplings for von Neumann algebras. Preprint (to be published in the proceedings of the Rome year on Quantum Probability 1986/1987).

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  6. A. Connes: Sur le theoreme de Radon-Nikodym pour les poids normale fideles semifinis. Bull. Sci. Math. Sec. II, 97 (1973), 253–258.

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  7. B. Kümmerer: Survey on a theorey of non-commutative stationary Markov processes. In Quantum Probability and Applications III, Proceedings Oberwolfach 1987, (L.Accardi, W.v.Waldenfels, Eds.), Springer Lecture Notes in Mathematics 1303 (1988), 154–182.

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© 1990 Springer-Verlag

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Cecchini, C., Kümmerer, B. (1990). Stochastic transitions on preduals of von Neumann algebras. In: Accardi, L., von Waldenfels, W. (eds) Quantum Probability and Applications V. Lecture Notes in Mathematics, vol 1442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085505

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  • DOI: https://doi.org/10.1007/BFb0085505

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53026-8

  • Online ISBN: 978-3-540-46311-5

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