Skip to main content
Log in

Local Ergodic Theorems in Symmetric Spaces of Measurable Operators

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

Local mean and individual (with respect to almost uniform convergence in Egorov’s sense) ergodic theorems are established for actions of the semigroup \({\mathbb {R}}_+^d\) in symmetric spaces of measurable operators associated with a semifinite von Neumann algebra.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Brunel, A.: Theóremè ergodique ponctuel pour un semigroupe commutatif finiment engendré de contractions de \(L^1\). AIHP B 9, 327–343 (1973)

    MATH  Google Scholar 

  2. Castrandas, E.: A local ergodic theorem in semifinite von Neumann algebras. Algebras Groups Geom. 13, 71–80 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Chilin, V., Litvinov, S., Skalski, A.: A few remarks in non-commutative ergodic theory. J. Oper. Theory 53(2), 331–350 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Chilin, V., Litvinov, S.: Ergodic theorems in fully symmetric spaces of \(\tau \)-measurable operators. Stud. Math. 288(2), 177–195 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chilin, V., Litvinov, S.: Individual ergodic theorems in noncommutative Orlicz spaces. Positivity 21(1), 49–59 (2017)

    Article  MathSciNet  Google Scholar 

  6. Chilin, V., Litvinov, S.: Individual ergodic theorems for semifinite von Neumann algebras. arXiv:1607.03452 [math.OA]

  7. Chilin, V., Sukochev, F.: Weak convergence in non-commutative symmetric spaces. J. Oper. Theory 31, 35–55 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Conze, J.P., Dang-Ngoc, N.: Ergodic theorems for noncommutative dynamical systems. Invent. Math. 46, 1–15 (1978)

    Article  MathSciNet  Google Scholar 

  9. Dodds, P.G., Dodds, T.K., Pagter, B.: Fully symmetric operator spaces. Integral Equ. Oper. Theory 15, 942–972 (1992)

    Article  MathSciNet  Google Scholar 

  10. Dodds, P.G., Dodds, T.K., Pagter, B.: Noncommutative Köthe duality. Trans. Am. Math. Soc. 339(2), 717–750 (1993)

    MathSciNet  MATH  Google Scholar 

  11. Edgar, G.A., Sucheston, L.: Stopping Times and Directed Processes. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  12. Fack, T., Kosaki, H.: Generalized \(s\)-numbers of \(\tau \)-measurable operators. Pac. J. Math. 123, 269–300 (1986)

    Article  MathSciNet  Google Scholar 

  13. Junge, M., Xu, Q.: Noncommutative maximal ergodic theorems. J. Am. Math. Soc. 20(2), 385–439 (2007)

    Article  MathSciNet  Google Scholar 

  14. Krein, S.G., Petunin, J.I., Semenov, E.M.: Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54. American Mathematical Society, Providence (1982)

  15. Kalton, N.J., Sukochev, F.A.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Krengel, U.: Ergodic Theorems. Walter de Gruyer, Berlin (1985)

    Book  Google Scholar 

  17. Litvinov, S.: Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems. Proc. Am. Math. Soc. 140(7), 2401–2409 (2012)

    Article  MathSciNet  Google Scholar 

  18. Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)

    Article  MathSciNet  Google Scholar 

  19. Pisier, G., Xu, Q.: Noncommutative \(L^p\)-spaces. Handb. Geom. Banach Spaces 2, 1459–1517 (2003)

    Article  Google Scholar 

  20. Rubshtein, B.A., Muratov, M.A., Grabarnik, G.Y., Pashkova, Y.S.: Foundations of Symmetric Spaces of Measurable Functions. Lorentz, Marcinkiewicz and Orlicz Spaces. Developments in Mathematics, vol. 45, Springer, Basel(2016)

    Chapter  Google Scholar 

  21. Watanabe, S.: Ergodic theorems for dynamical semi-groups on operator algebras. Hokkaido Math. J. 8, 176–190 (1979)

    Article  MathSciNet  Google Scholar 

  22. Yeadon, F.J.: Non-commutative \(L^p\)-spaces. Math. Proc. Camb. Philos. Soc. 77, 91–102 (1975)

    Article  MathSciNet  Google Scholar 

  23. Yeadon, F.J.: Ergodic theorems for semifinite von Neumann algebras. I. J. Lond. Math. Soc. 16(2), 326–332 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Semyon Litvinov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chilin, V., Litvinov, S. Local Ergodic Theorems in Symmetric Spaces of Measurable Operators. Integr. Equ. Oper. Theory 91, 15 (2019). https://doi.org/10.1007/s00020-019-2519-1

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-019-2519-1

Mathematics Subject Classification

Keywords

Navigation