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Quantum diffusions on the algebra of all bounded operators on a hilbert space

Part of the Lecture Notes in Mathematics book series (LNM,volume 1396)

Abstract

Quantum diffusions on the algebra B(ℋ0) of all bounded operators on a Hilbert space are analysed. When ℋ0 is finite dimensional all such diffusions are described by unitary processes. When ℋ0 is infinite dimensional the general such diffusion is shown to be a unitary perturbation, which can be constructed explicitly, of a class of quantum diffusions completely characterised by an endomorphism of B(ℋ0).

Keywords

  • Unitary Process
  • Stochastic Differential Equation
  • Quantum Diffusion
  • Commun Math Phys
  • Fundamental Formula

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Work partly supported by SERC grant GR/D51292 and completed when the author was visiting the University of Strasbourg, whose hospitality is gratefully acknowledged.

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References

  1. M Evans, Existence of quantum diffusions, submitted to Modern Probability and its Applications.

    Google Scholar 

  2. M Evans and R L Hudson, Multidimensional quantum diffusions, in Quantum Probability III, Oberwolfach (1987) Proceedings, ed L Accardi and W von Waldenfels, Springer LNM 1303 (1988).

    Google Scholar 

  3. R L Hudson, Algebraic theory of quantum diffusions, to appear in Swansea (1986) Proceedings, ed A Truman.

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  4. R L Hudson, Quantum diffusions and cohomology of algebras, in Proceedings of First World Congress of Bernoulli Society, Vol 1, ed Yu Prohorov and V V Sazonov, pp 479–485 (1987).

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  5. R L Hudson and K R Parthasarathy, Quantum Ito's formula and stochastic evolutions, Commun Math Phys, 93, 301–323 (1984).

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. R V Kadison and J R Ringrose, Fundamentals of the theory of operator algebras Volume 1, Academic Press (1983), Volume 2, Academic Press (1986).

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  7. R V Kadison and J R Ringrose, Cohomology of operator algebras I, Acta Math, 126, 227–243 (1971).

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1989 Springer-Verlag

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Hudson, R. (1989). Quantum diffusions on the algebra of all bounded operators on a hilbert space. In: Accardi, L., von Waldenfels, W. (eds) Quantum Probability and Applications IV. Lecture Notes in Mathematics, vol 1396. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083556

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  • DOI: https://doi.org/10.1007/BFb0083556

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51613-2

  • Online ISBN: 978-3-540-46713-7

  • eBook Packages: Springer Book Archive