Skip to main content
SpringerLink
Log in

On spectral gaps of Markov maps

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

It is shown that if a Markov map T on a noncommutative probability space M has a spectral gap on L2(M), then it also has one on Lp(M) for 1 < p < ∞. For fixed p, the converse also holds if T is factorizable. Some results are also new for classical probability spaces.

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. C. Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probability Theory and Related Fields 135 (2006), 520–546.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Caspers, J. Parcet, M. Perrin and É. Ricard, Noncommutative de Leeuw theorems, Forum of Mathematics. Sigma 3 (2015), e21, 59 pp.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. D. Choi and A Schwarz, inequality for positive linear maps on C*-algebras, Illinois Journal of Mathematics 18 (1974), 565–574.

    MathSciNet  MATH  Google Scholar 

  4. J. M. Conde-Alonso, T. Mei and J. Parcet, Large BMO spaces vs interpolation, Analysis & PDE 8 (2015), 713–746.

    Article  MathSciNet  MATH  Google Scholar 

  5. U. Haagerup and M. Musat, Factorization and dilation problems for completely positive maps on von Neumann algebras, Communications in Mathematical Physics 303 (2011), 555–594.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Heilman, E. Mossel and K. Oleszkiewicz, Strong contraction and influences in tail spaces, Transactions of the American Mathematical Society 369 (2017), 4843–4863.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Pisier and Q. Xu, Non-commutative Lp-spaces, in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.

    Chapter  Google Scholar 

  8. É. Ricard, Hölder estimates for noncommutative Mazur maps, Archiv der Mathematik 104 (2015), 37–45.

    Article  MathSciNet  MATH  Google Scholar 

  9. É. Ricard, An inequality in noncommutative Lp-spaces, Journal of Mathematical Analysis and Applications 438 (2016), 53–58.

    Article  MathSciNet  MATH  Google Scholar 

  10. É. Ricard, A Markov dilation for self-adjoint Schur multipliers, Proceedings of the American Mathematical Society 136 (2008), 4365–4372.

    Article  MathSciNet  MATH  Google Scholar 

  11. É. Ricard and Q. Xu, A noncommutative martingale convexity inequality, Annals of Probability 44 (2016), 867–882.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193, Barcelona, Spain

    José M. Conde-Alonso

  2. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 13–15, 28049, Madrid, Spain

    Javier Parcet

  3. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie, 14032, Caen Cedex, France

    Éric Ricard

Authors
  1. José M. Conde-Alonso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Javier Parcet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Éric Ricard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José M. Conde-Alonso.

Additional information

J. M. Conde-Alonso was supported in part by ERC Grant 32501.

J. Parcet was supported in part by CSIC Grant PIE 201650E030 (Spain). Received February 1, 2017 and in revised form April 19, 2017

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Conde-Alonso, J.M., Parcet, J. & Ricard, É. On spectral gaps of Markov maps. Isr. J. Math. 226, 189–203 (2018). https://doi.org/10.1007/s11856-018-1693-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1693-1

Search

Navigation