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Stability of diametral diameter two properties

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Abstract

We prove that the diametral diameter two properties are inherited by F-ideals (e.g., M-ideals). On the other hand, these properties are lifted from an M-ideal to the superspace under strong geometric assumptions. We also show that all of the diametral diameter two properties are stable under the formation of corresponding Köthe–Bochner spaces (e.g., \(L_p\)-Bochner spaces). Finally, we investigate when the projective tensor product of two Banach spaces has some diametral diameter two property.

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References

  1. Abrahamsen, T.A., Hájek, P., Nygaard, O., Talponen, J., Troyanski, S.: Diameter 2 properties and convexity. Studia Math. 232, 227–242 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Abrahamsen, T.A., Haller, R., Lima, V., Pirk, K.: Delta- and Daugavet points in Banach spaces. Proc. Edinb. Math. Soc. II. Ser. 63, 475–496 (2020)

    Article  MathSciNet  Google Scholar 

  3. Abrahamsen, T.A., Lima, V., Martiny, A., Troyanski, S.: Daugavet- and delta-points in Banach spaces with unconditional bases, preprint (2020), https://arxiv.org/abs/2007.04946

  4. Abrahamsen, T.A., Lima, V., Nygaard, O.: Remarks on diameter 2 properties. J. Convex Anal. 20, 439–452 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Acosta, M.D., Becerra-Guerrero, J., López-Pérez, G.: Stability results of diameter two properties. J. Convex Anal. 22, 1–17 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Avilés, A., Martínez-Cervantes, G.: Complete metric spaces with property \((Z)\) are length spaces. J. Math. Anal. Appl. 473, 334–344 (2019)

    Article  MathSciNet  Google Scholar 

  7. Becerra Guerrero, J., López-Pérez, G.: Relatively weakly open subsets of the unit ball in functions spaces. J. Math. Anal. Appl. 315(2), 544–554 (2006)

    Article  MathSciNet  Google Scholar 

  8. Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Big slices versus big relatively weakly open subsets in Banach spaces. J. Math. Anal. Appl. 428, 855–865 (2015)

    Article  MathSciNet  Google Scholar 

  9. Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Diametral diameter two properties in Banach spaces. J. Convex Anal. 25, 817–840 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Octahedrality in Lipschitz-free Banach spaces. Proc. R. Soc. Edinb. Sect. A Math. 148, 447–460 (2018)

    Article  MathSciNet  Google Scholar 

  11. Bonsall, F.F., Duncan, J.: Numerical Ranges II, vol. 10. Cambridge University Press, Cambridge. London Mathematical Society, London (1973)

  12. Daugavet, I.K.: A property of complete continuous operators in the space \(C\). Usp. Mat. Nauk 18, 157–158 (1963)

    MATH  Google Scholar 

  13. Haller, R., Langemets, J.: Two remarks on diameter 2 properties. Proc. Est. Acad. Sci. 63, 2–7 (2014)

    Article  Google Scholar 

  14. Haller, R., Langemets, J., Lima, V., Nadel, R.: Symmetric strong diameter two property. Mediterr. J. Math. 16, (2019)

  15. Haller, R., Langemets, J., Lima, V., Nadel, R., Rueda Zoca, A.: On Daugavet indices of thickness, J. Func. Anal (2020) in press

  16. Hardtke, J.-D.: On certain geometric properties in Banach spaces of vector-valued functions. Zh. Mat. Fiz. Anal. Geom. 16, 119–137 (2020)

    Article  MathSciNet  Google Scholar 

  17. Harmand, P., Werner, D., Werner, W.: \(M\)-ideals in Banach Spaces and Banach Algebras, vol. 1547. Springer-Verlag, Berlin (1993)

    Book  Google Scholar 

  18. Ivakhno, Y., Kadets, V.: Unconditional sums of spaces with bad projections. Visn. Khark. Univ. Ser. Mat. Prykl. Mat. Mekh. 645, 30–35 (2004)

    MATH  Google Scholar 

  19. Kadets, V.M., Shvidkoy, R.V., Sirotkin, G.G., Werner, D.: Banach spaces with the Daugavet property. Trans. Am. Math. Soc. 352, 855–873 (2000)

    Article  MathSciNet  Google Scholar 

  20. Lin, P.K.: Köthe-Bochner Function Spaces. Birkhäuser, Boston (2004)

    Book  Google Scholar 

  21. Martin, M., Zoca, A.R.: Daugavet property in projective symmetric tensor products of Banach spaces, preprint (2020), https://arxiv.org/abs/2010.15936

  22. Pirk, K.: Diametral diameter two properties, Daugavet- and\(\Delta \)-points in Banach spaces, Dissertationes Mathematicae Universitatis Tartuensis 133 (2020), https://dspace.ut.ee/handle/10062/68458

  23. Rueda Zoca, A., Tradacete, P., Villanueva, I.: Daugavet property in tensor product spaces, J. Inst. Math. Jussieu, pp. 1-20 (2019)

  24. Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer, London (2002)

    Book  Google Scholar 

  25. Shvydkoy, R.V.: Geometric aspects of the Daugavet property. J. Funct. Anal. 176, 198–212 (2000)

    Article  MathSciNet  Google Scholar 

  26. Werner, D.: \(M\)-ideals and the “basic inequality.” J. Approx. Theory 76, 21–30 (1994)

  27. Werner, D.: Recent progress on the Daugavet property, Ir. Math. Soc. Bull. 46 (2001), pp. (2001), 77–97

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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 the recent paper [16].

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

    Johann Langemets & Katriin Pirk

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  1. Johann Langemets
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  2. Katriin Pirk
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Correspondence to Katriin Pirk.

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

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Langemets, J., Pirk, K. Stability of diametral diameter two properties. RACSAM 115, 96 (2021). https://doi.org/10.1007/s13398-021-01038-y

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  • DOI: https://doi.org/10.1007/s13398-021-01038-y

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