Skip to main content

Conditional expectation in quantum probability

  • Conference paper
  • First Online:
Quantum Probability and Applications III

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1303))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Accardi, C. Cecchini, Conditinal expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal.,45(1982), 245–273.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Accardi, C. Cecchini, Surjectivity of the conditional expectation on the L1-spaces, Lecture Notes in Math., 992, 436–442, Springer, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra, Publ. Res. Inst. Math. Sci.,9(1973), 165–209.

    Article  MathSciNet  MATH  Google Scholar 

  4. O.Bratteli, D.W.Robinson, Operator algebras and quantum statistical mechanics I,II, Springer 1979 and 1981.

    Google Scholar 

  5. C. Cecchini, D. Petz, Norm convergence of generalized martingales in LP-spaces over von Neumann algebras, Acta Sci. Math., 48(1985), 55–63.

    MathSciNet  MATH  Google Scholar 

  6. C.Cecchini, Some non-commutative Radon-Nikodym theorems, in this volume.

    Google Scholar 

  7. A. Connes, Caracterization des espaces vectoriels ordonnées sousjacents aux algebres de von Neumann, Ann. Inst. Fourier, 24(1974), 121–155.

    Article  MathSciNet  Google Scholar 

  8. S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 75(1984), 493–536.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Frigerio, Duality of completely positive quasi-free maps and a theorem of L.Accardi and C.Cecchini, Boll. Un. Mat. Ital., 6(1983), 269–281.

    MathSciNet  MATH  Google Scholar 

  10. F. Hiai, M. Ohya, M. Tsukada, Sufficiency, KMS condition and relative entropy in von Neumann algebras, Pacific J. Math., 96(1981), 99–109.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Hiai, M. Ohya, M. Tsukada, Sufficiency and relative entropy in *-algebras with applications to quantum systems, Pacific J. Math., 107(1983), 117–140.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Hiai, M. Tsukada, Strong martingale convergence of generalized conditional expectations on von Neumann algebras, Trans. Amer. Math. Soc., 282(1984), 791–798.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Kümmerer, Adjoints of operators on W*-algebras, Preprint, Tübingen, 1985.

    MATH  Google Scholar 

  14. R. Longo, Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard split inclusions, Invent. Math., 76(1984), 145–155.

    Article  MathSciNet  MATH  Google Scholar 

  15. S.Ch. Moy, Characterization of conditional expectation as a transform of function spaces, Pacific J. Math., 4(1954), 47–63.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Nakamura, T. Turumaru, Expectations in an operator algebra, Tohoku J. Math., 6(1954), 189–204

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Petz, Quantum ergodic theorems, Proceedings of the Workshop on Quantum Probability, Lecture Notes in Math., 1055, 289–300, Springer, 1984.

    Article  MathSciNet  Google Scholar 

  18. D. Petz, A dual in von Neumann algebras, Quart. J. Math. Oxford, 35(1984), 475–483.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys., 105(1986), 123–131.

    Article  MathSciNet  MATH  Google Scholar 

  20. D.Petz, Sufficiency of channels over von Neumann algebras, Quart. J. Math. Oxford, to appear.

    Google Scholar 

  21. G.A. Raggio, Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of a von Neumann algebra, Lett. Math. Phys., 6(1982), 233–236.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Riesz, B.Sz. Nagy, Lectures on functional analysis, Ungar, New York, 1955.

    MATH  Google Scholar 

  23. M. Takesaki, Conditional expectations in von Neumann algebras, J.Funct. Analysis, 9(1972), 306–321.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Umegaki, Conditional expectations in an operator algebra, Tohoku J. Math., 6(1954), 177–181.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Umegaki, Conditional expectations in an operator algebra IV (entropy and information), Kodai Math. Sem. Rep., 14(1962), 59–85.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Luigi Accardi Wilhelm von Waldenfels

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Petz, D. (1988). Conditional expectation in quantum probability. In: Accardi, L., von Waldenfels, W. (eds) Quantum Probability and Applications III. Lecture Notes in Mathematics, vol 1303. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078067

Download citation

  • DOI: https://doi.org/10.1007/BFb0078067

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18919-0

  • Online ISBN: 978-3-540-38846-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics