Skip to main content
SpringerLink
Log in

Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group S on the straight line is introduced. The class of cocycle perturbations of S by unitary Markovian cocycles W with the property W t − I ∈ S 2 (the Hilbert—Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. G. Amosov, “Cocycle perturbation of quasifree algebraic K-flow leads to required asymptotic dynamics of associated completely positive semigroup,” Infin. Dimen. Anal. Quantum Probab. Rel. Top., 3 (2000), no. 2, 237–246.

    MATH  MathSciNet  Google Scholar 

  2. G. G. Amosov, “Stationary quantum stochastic processes from cohomological point of view,” in: Quantum Probability and Infinite-Dimensional Analysis (Burg, 2001) (W. Freudenberg, Ed.), Quantum Probab. White Noise Anal., vol. 15, World Sci., River Edge, NJ, 2003, pp. 29–40.

    Google Scholar 

  3. G. G. Amosov, “On Markovian cocycle perturbations in classical and quantum probability,” Int. J. Math. Math. Sci. (2003), no. 54, 3443–3467.

  4. G. G. Amosov, “On Markovian perturbations of the group of unitary operators associated with a random process with stationary increments,” Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.], 49 (2004), 145–155.

    MATH  MathSciNet  Google Scholar 

  5. L. Accardi, A. Frigerio, and J. T. Lewis, “Quantum stochastic processes,” Publ. Res. Inst. Math. Sci., 18 (1982), no. 1, 97–133.

    MathSciNet  Google Scholar 

  6. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Monographs in Math., vol. 85, Birkhauser, Basel, 1992.

    Google Scholar 

  7. J. M. Lindsay and S. J. Wills, “Markovian cocycles on operator algebras adapted to a Fock filtration,” J. Funct. Anal., 178 (2000), 269–300.

    Article  MathSciNet  Google Scholar 

  8. V. Liebscher, “How to generate Markovian cocycles on boson Fock space,” Infin. Dimen. Anal. Quantum Probab. Rel. Top., 4 (2001), 215–219.

    MATH  MathSciNet  Google Scholar 

  9. B. V. R. Bhat, “Minimal isometric dilations of operator cocycles,” Integral Equations Operator Theory, 42 (2002), no. 2, 125–141.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. G. Amosov, “On cocycle conjugacy of quasifree endomorphism semigroups on the CAR algebra,” J. Math. Sci., 105 (2001), 2496–2503.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. V. Bulinskii, “Algebraic K-flows and semiflows of Powers shifts,” Uspekhi Mat. Nauk [Russian Math. Surveys], 51 (1996), 145–146.

    MATH  MathSciNet  Google Scholar 

  12. R. T. Powers, “An index theory for semigroups of *-endomorphisms of \(\mathcal{B}(\mathcal{H})\) and types II1 factors,” Canad. J. Math., 40 (1988), no. 1, 86–114.

    MATH  MathSciNet  Google Scholar 

  13. W. Arveson, “Continuous analogues of Fock space,” Mem. Amer. Math. Soc., 80 (1989), no. 409, 1–66.

    MathSciNet  Google Scholar 

  14. B. V. R. Bhat and M. Skeide, “Tensor product systems of Hilbert modules and dilations of completely positive semigroups,” Infin. Dimen. Anal. Quantum Probab. Rel. Top., 3 (2000), 519–575.

    MathSciNet  Google Scholar 

  15. S. D. Barreto, B. V. R. Bhat, V. Liebscher, and M. Skeide, “Type I product systems of Hilbert modules,” J. Funct. Anal., 212 (2004), 121–181.

    Article  MathSciNet  Google Scholar 

  16. L. Accardi, G. G. Amosov, and U. Franz, “Second quantized automorphisms of the renormalized square of white noise (RSWN) algebra,” Infin. Dimen. Anal. Quantum Probab. Rel. Top., 7 (2004), 183–194.

    MathSciNet  Google Scholar 

  17. N. K. Nikol'skii, Lectures on the Shift Operator [in Russian], Nauka, Moscow, 1980.

    Google Scholar 

  18. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Space, North-Holland, Amsterdam, 1970; Russian translation: Mir, Moscow, 1970.

    Google Scholar 

  19. G. G. Amosov and A. D. Baranov, “On perturbations of the group of shifts on the line by unitary cocycles,” Proc. Amer. Math. Soc., 132 (2004), 3269–3273.

    Article  MathSciNet  Google Scholar 

  20. A. D. Baranov, “Isometric embeddings of the spaces K⊖ in the upper half-plane,” Problemy Mat. Anal., 21 (2000), 30–44.

    MATH  Google Scholar 

  21. P. R. Ahern and D. N. Clark, “On functions orthogonal to invariant subspaces,” Acta Math., 124 (1970), 191–204.

    MathSciNet  Google Scholar 

  22. H. Araki, “On quasifree states of CAR and Bogoliubov automorphisms,” Contemp. Math., 62 (1985), 21–141.

    Google Scholar 

  23. G. G. Emch, “Generalized K-flows,” Commun. Math. Phys., 49 (1976), 191–215.

    Article  MATH  MathSciNet  Google Scholar 

  24. D. Evans, “Completely positive quasifree maps on the CAR algebra,” Commun. Math. Phys., 70 (1979), 53–68.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Moscow, Russia

    G. G. Amosov

  2. St. Petersburg State University, St. Petersburg, Russia

    A. D. Baranov

Authors
  1. G. G. Amosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. D. Baranov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Matematicheskie Zametki, vol. 79, no. 1, 2006, pp. 3–18.

Original Russian Text Copyright © 2006 by G. G. Amosov, A. D. Baranov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Amosov, G.G., Baranov, A.D. Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line. Math Notes 79, 3–17 (2006). https://doi.org/10.1007/s11006-006-0001-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11006-006-0001-2

Key words

Search

Navigation