Skip to main content
Log in

A-ergodicity of probability measures on locally compact groups

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

Let G be a locally compact group with the left Haar measure \(m_{G}\) and let \(A=\left[ a_{n,k}\right] _{n,k=0}^{\infty }\) be a strongly regular matrix. We show that if \(\mu \) is a power bounded measure on G, then there exists an idempotent measure \(\theta _{\mu }\) such that

$$\begin{aligned} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}=\theta _{\mu }. \end{aligned}$$

If \(\mu \) is a probability measure on a compact group G,  then

$$\begin{aligned} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}=\overline{m}_{H}, \end{aligned}$$

where H is the closed subgroup of G generated by \(\text{ supp }\mu \) and \( \overline{m}_{H}\) is the measure on G defined by \(\overline{m}_{H}\left( E\right) :=m_{H}\left( E\cap H\right) \) for every Borel subset E of G.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /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. Derriennic, Y., Lin, M.: Convergence of iterates of averages of certain operator representations and of convolution powers. J. Funct. Anal. 85, 86–102 (1989)

    Article  MathSciNet  Google Scholar 

  2. Dixmier, J.: Les \(C^{\ast }\)-algèbres et leurs représentations. Fasc. XXIX. Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, Cahiers Scientifiques (1964)

    Google Scholar 

  3. Galindo, J., Jorda, E.: Ergodic properties of convolution operators. J. Oper. Theory 86, 469–501 (2021)

    Article  MathSciNet  Google Scholar 

  4. Grenander, U.: Probabilities on Algebraic Structures. Second edition. Almqvist & Wiksell, Stockholm; John Wiley & Sons, Inc., New York-London (1968)

  5. Kawada, Y., Itô, K.: On the probability distribution on a compact group. Proc. Phys.-Math. Soc. Japan 22, 977–998 (1940)

    MathSciNet  Google Scholar 

  6. Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin, New York (1985)

    Book  Google Scholar 

  7. Lyubich, Y.I.: Introduction to the Theory of Banach Representations of Groups. Translated from the Russian by A. Jacob. Operator Theory: Advances and Applications, 30. Birkhäuser, Basel (1988)

  8. Mustafayev, H.: Mean ergodic theorems for multipliers on Banach algebras. J. Fourier Anal. Appl. 25, 393–426 (2019)

    Article  MathSciNet  Google Scholar 

  9. Mustafayev, H.: A note on the Kawada-Itô theorem. Statist. Probab. Lett. 181, Paper No. 109261, 6 pp. (2022)

  10. Neufang, M., Salmi, P., Skalski, A., Spronk, N.: Fixed points and limits of convolution powers of contractive quantum measures. Indiana Univ. Math. J. 70, 1971–2009 (2021)

    Article  MathSciNet  Google Scholar 

  11. Petersen, G.M.: Regular Matrix Transformations. McGraw-Hill Publishing Co., Ltd., London-New York-Toronto (1966)

    Google Scholar 

  12. Shapiro, J.H.: Every composition operator is (mean) asymptotically Toeplitz. J. Math. Anal. Appl. 333, 523–529 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referee for his helpful remarks and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Heybetkulu Mustafayev.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mustafayev, H. A-ergodicity of probability measures on locally compact groups. Arch. Math. 122, 47–57 (2024). https://doi.org/10.1007/s00013-023-01938-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-023-01938-y

Keywords

Mathematics Subject Classification

Navigation