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  • Johann Langemets (a1) and Ginés López-Pérez (a2)


We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.



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The work of J. Langemets was supported by the Estonian Research Council grant (PUTJD702), by institutional research funding IUT (IUT20-57) of the Estonian Ministry of Education and Research, and by a grant of the Institute of Mathematics of the University of Granada (IEMath-GR). The work of G. López-Pérez was supported by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE) and by Junta de Andalucía Grant FQM-0185.



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1. Becerra Guerrero, J., López-Pérez, G. and Rueda Zoca, A., Octahedral norms and convex combination of slices in Banach spaces, J. Funct. Anal. 266 (2014), 24242435.
2. Becerra Guerrero, J., López-Pérez, G. and Rueda Zoca, A., Extreme differences between weakly open subsets and convex combinations of slices in Banach spaces, Adv. Math. 269 (2015), 5670.
3. Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64 (Longman Scientific & Technical, Harlow, 1993).
4. Dilworth, S. J., Girardi, M. and Hagler, J., Dual Banach spaces which contain isometric copy of L 1 , Bull. Pol. Acad. Sci. Math. 48 (2000), 112.
5. Ghoussoub, N., Godefroy, G., Maurey, B. and Schachermayer, W., Some topological and geometrical structures in Banach spaces, Mem. Amer. Math. Soc. 378 (1987), 116.
6. Godefroy, G., Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 115.
7. Godefroy, G. and Kalton, N. J., The ball topology and its applications, Contemp. Math. 85 (1989), 195237.
8. Godefroy, G. and Maurey, B., Normes lisses et anguleuses sur les espaces de Banach séparables, unpublished preprint.
9. Hagler, J. and Stegall, C., Banach spaces whose duals contain complemented subspaces isomorphic to C[0, 1] , J. Funct. Anal. 13 (1973), 233251.
10. Haller, R., Langemets, J. and Põldvere, M., On duality of diameter 2 properties, J. Conv. Anal. 22(2) (2015), 465483.
11. Kadets, V., Shepelska, V. and Werner, D., Thickness of the unit sphere, 1 -types, and the almost Daugavet property, Houston J. Math. 37 (2011), 867878.
12. Kadets, V., Shvidkoy, R., Sirotkin, G. and Werner, D., Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), 855873.
13. Kaijser, S., A note on dual Banach spaces, Math. Scand. 41 (1977), 325330.
14. López-Pérez, G., Martín, M. and Rueda Zoca, A., Strong diameter two property and convex combination of slices reaching the unit sphere, Mediterr. J. Maths. (to appear), Preprint, 2017, arXiv:1703.04749.
15. Maurey, B., Types and 1 -subspaces, in Texas Functional Analysis Seminar, Austin, Texas 1982/1983, Longhorn Notes.
16. Rosenthal, H., A characterization of Banach spaces containing l 1 , Proc. Natl. Acad. Sci. USA 71 (1974), 24112413.
17. Schachermayer, W., Sersouri, A. and Werner, E., Moduli of nondentability and the Radon–Nikodým property in Banach spaces, Israel J. Math. 65 (1989), 225257.
18. Yagoub-Zidi, Y., Some isometric properties of subspaces of function spaces, Mediterr. J. Math. 10(4) (2013), 19051915.
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  • Johann Langemets (a1) and Ginés López-Pérez (a2)


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