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A characterization of Banach spaces containing ℓ1(κ) via ball-covering properties

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Abstract

In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of ℓ1 if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy’s result to higher cardinalities. We prove that, for a regular cardinal κ, a Banach space X contains an isomorphic copy of ℓ1(κ) if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by strictly less than κ many open balls not containing αBX, where α ∊ (0, 1). We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.

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References

  1. A. Avilés, S. Ciaci, J. Langemets, A. Lissitsin and A. Rueda Zoca, Transfinite almost square Banach spaces, https://arxiv.org/abs/2204.13449.

  2. A. Avilés and G. Martínez-Cervantes, Complete metric spaces with property (Z) are length spaces, Journal of Mathematical Analysis and Applications 473 (2019), 334–344.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Avilés, G. Martínez-Cervantes and A. Rueda Zoca, A renorming characterization of Banach spaces containing ℓ 1(κ), Publicacions Matemàtiques, to appear, https://arxiv.org/abs/2104.13858.

  4. F. F. Bonsall and J. Duncan, Numerical Ranges II, London Mathematical Society Lecture Note Series, Vol. 10, Cambridge University Press, New York-London, 1973.

  5. T. Bosenko and V. Kadets, Daugavet centers, Journal of Mathematical Physics, Analysis, Geometry 6 (2010), 3–20.

    MathSciNet  MATH  Google Scholar 

  6. L. Cheng, Ball-covering property of Banach spaces, Israel Journal of Mathematics 156 (2006), 111–123.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Cheng, Q. Cheng and X. Liu, Ball-covering property of Banach spaces that is not preserved under linear isomorphism, Science in China. Series A. Mathematics 51 (2008), 143–147.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Cheng, Q. Cheng and H. Shi, Minimal ball-coverings in Banach spaces and their application, Studia Mathematica 192 (2009), 15–27.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Cheng, V. Kadets, B. Wang and W. Zhang, A note on ball-covering property of Banach spaces, Journal of Mathematical Analysis and Applications 371 (2010), 249–253.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Cheng, H. Shi and W. Zhang, Every Banach space with a w*-separable dual has a 1 + ε-equivalent norm with the ball covering property, Science in China. Series A. Mathematics 52 (2009), 1869–1874.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. K. Daugavet, A property of completely continuous operators in the space C, Uspekhi Matematicheskikh Nauk 18 (1963), 157–158.

    MathSciNet  MATH  Google Scholar 

  12. V. P. Fonf and C. Zanco, Covering spheres of Banach spaces by balls, Mathematische Annalen 344 (2009), 939–945.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. García-Lirola, A. Procházka and A. Rueda Zoca, A characterisation of the Daugavet property in spaces of Lipschitz functions, Journal of Mathematical Analysis and Applications 464 (2018), 473–492.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Godefroy, Metric characterization of first Baire class linear forms and octahedral norms, Studia Mathematica 95 (1989), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Godefroy and N. J. Kalton, The ball topology and its applications. in Banach Space Theory (Iowa City, IA, 1987), Contemporary Mathematics, Vol. 85, American Mathematical Society, Providence, RI, 1989, pp. 195–237.

    Chapter  Google Scholar 

  16. G. Godefroy and B. Maurey, Normes lisses et anguleuses sur les espaces de Banach séparables, unpublished preprint.

  17. A. J. Guirao, A. Lissitsin and V. Montesinos, Some remarks on the ball-covering property, Journal of Mathematical Analysis and Applications 479 (2019), 608–620.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Hájek, V. Montesinos, J. Vanderwerff and V. Zizler, Biorthogonal Systems in Banach Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Vol. 26, Springer, New York, 2008.

  19. R. Haller and J. Langemets, Geometry of Banach spaces with an octahedral norm, Acta et Commentationes Universitatis Tartuensis de Mathematica 18 (2014), 125–133.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Haller, J. Langemets and M. Põldvere, On duality of diameter 2 properties, Journal of Convex Analysis 22 (2015), 465–483.

    MathSciNet  MATH  Google Scholar 

  21. V. Kadets, V. Shepelska and D. Werner, Thickness of the unit sphere, ℓ 1-types, and the almost Daugavet property, Houston Journal of Mathematics 37 (2011), 867–878.

    MathSciNet  MATH  Google Scholar 

  22. V. M. Kadets, R. V. Shvidkoy, G. G. Sirotkin and D. Werner, Banach spaces with the Daugavet property, Transactions of the American Mathematical Society 352 (2000), 855–873.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Langemets and G. López-Pérez, Bidual octahedral renormings and strong regularity in Banach spaces, Journal of the Institute of Mathematics of Jussieu 20 (2021), 569–585.

    Article  MathSciNet  MATH  Google Scholar 

  24. Z. Luo and B. Zheng, Stability of the ball-covering property, Studia Mathematica 250 (2020), 19–34.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. L. Papini, Covering the sphere and the ball in Banach spaces, Communications in Applied Analysis 13 (2009), 579–586.

    MathSciNet  MATH  Google Scholar 

  26. H. P. Rosenthal, A characterization of Banach spaces containing l 1, Proceedings of the National Academy of Sciences of the United States of America 71 (1974), 2411–2413.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Werner, Recent progress on the Daugavet property, Irish Mathematical Society Bulletin 46 (2001), 77–97.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for a careful reading of the manuscript and for suggestions that improved the exposition, in particular, for pointing out a better statement of Theorem 3.1.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Tartu Narva mnt 18, 51009, Tartu, Estonia

    Stefano Ciaci, Johann Langemets & Aleksei Lissitsin

Authors
  1. Stefano Ciaci
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  2. Johann Langemets
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  3. Aleksei Lissitsin
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Corresponding author

Correspondence to Stefano Ciaci.

Additional information

This work was supported by the Estonian Research Council grants (PSG487) and (PRG877).

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Ciaci, S., Langemets, J. & Lissitsin, A. A characterization of Banach spaces containing ℓ1(κ) via ball-covering properties. Isr. J. Math. 253, 359–379 (2023). https://doi.org/10.1007/s11856-022-2363-x

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  • DOI: https://doi.org/10.1007/s11856-022-2363-x

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