Skip to main content

Examples of markov dilations over the 2×2 matrices

Part of the Lecture Notes in Mathematics book series (LNM,volume 1055)

Abstract

We construct and discuss some explicit examples of Markov dilations for semigroups of completely positive operators on the W*-algebra of 2×2 matrices. In particular, we obtain a continuous Markov dilation for a semigroup of non-quasifree operators.

Keywords

  • Conditional Expectation
  • Positive Operator
  • Markov Property
  • Discrete Dynamical System
  • Continuous Dynamical System

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.

This paper reports on results which are part of a research project supported by the Deutsche Forschungsgemeinschaft

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Accardi, L.; Frigerio, A.; Lewis, J.T.: Quantum Stochastic Processes. Publ. RIMS, Kyoto Univ. 18 (1982), 97–133.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Emch, G.G.; Albeverio, S.; Eckmann, J.-P.: Quasi-Free Generalized K-Flows. Rep. Math. Phys. 13 (1978), 73–85.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  3. Emch, G.G.; Varilly, J.C.: On the Standard Form of the Bloch Equation. Lett. Math. Phys. 3 (1979), 113–116.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  4. Evans, D.E.: Completely Positive Quasi-Free Maps on the CAR Algebra. Comm. Math. Phys. 70 (1979), 53–68.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  5. Hida, T.: Brownian Motion. Springer Verlag, Berlin-Heidelberg-New York 1980.

    CrossRef  MATH  Google Scholar 

  6. Kern, M.; Nagel, R.; Palm, G.: Dilations of Positive Operators: Construction and Ergodic Theory. Math. Z. 156 (1977), 256–277.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Kümmerer, B.: Markov Dilations of Completely Positive Operators on W*-Algebras. To appear in "Topics in Modern Operator Theory", Proc. Timisoara-Herculane 1982, Birkhäuser Verlag, Basel 1983.

    Google Scholar 

  8. Kümmerer, B.: A Dilation Theory for Completely Positive Operators on W*-Algebras. Thesis, Tübingen 1982.

    Google Scholar 

  9. Kümmerer, B.; Schröder, W.: A Markov Dilation of a Non-Quasifree Bloch Evolution. To appear in Comm. Math. Phys..

    Google Scholar 

  10. Kümmerer, B.; Schröder, W.: A Survey of Markov Dilations for the Spin-½-Relaxation and Physical Interpretation. Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 1981/82, 187–213.

    Google Scholar 

  11. Kümmerer, B.: A Non-Commutative Example of a Continuous Markov Dilation. Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 1982/83, 61–91.

    Google Scholar 

  12. Lewis, J.T.; Thomas, L.C.: How to Make a Heat Bath. In "Functional Integration", Proc. Cumberland Lodge, London, Clarendon Press, Oxford 1975, 97–123.

    Google Scholar 

  13. Sz.-Nagy, B.; Foias, C.: Harmonic Analysis of Operators on Hilbert Space. North Holland, Amsterdam 1970.

    MATH  Google Scholar 

  14. Varilly, J.C.: Dilation of a Non-Quasifree Dissipative Evolution. Lett. Math. Phys. 5 (1981), 113–116.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Kümmerer, B. (1984). Examples of markov dilations over the 2×2 matrices. In: Accardi, L., Frigerio, A., Gorini, V. (eds) Quantum Probability and Applications to the Quantum Theory of Irreversible Processes. Lecture Notes in Mathematics, vol 1055. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071725

Download citation

  • DOI: https://doi.org/10.1007/BFb0071725

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12915-8

  • Online ISBN: 978-3-540-38798-5

  • eBook Packages: Springer Book Archive