The diametral strong diameter 2 property of Banach spaces is the same as the Daugavet property
Author:
Vladimir Kadets
Journal:
Proc. Amer. Math. Soc. 149 (2021), 2579-2582
MSC (2020):
Primary 46B20
DOI:
https://doi.org/10.1090/proc/15448
Published electronically:
March 22, 2021
MathSciNet review:
4246808
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We demonstrate the result stated in the title, thus answering an open question asked by Julio Becerra Guerrero, Ginés López-Pérez and Abraham Rueda Zoca [J. Convex. Anal. 25, (2018), pp. 817–840].
- Antonio Avilés, Vladimir Kadets, Miguel Martín, Javier Merí, and Varvara Shepelska, Slicely countably determined Banach spaces, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4871–4900. MR 2645054, DOI https://doi.org/10.1090/S0002-9947-10-05038-5
- Julio Becerra Guerrero, Ginés López-Pérez, and Abraham Rueda Zoca, Diametral diameter two properties in Banach spaces, J. Convex Anal. 25 (2018), no. 3, 817–840. MR 3818544, DOI https://doi.org/10.1017/s0308210517000373
- I. K. Daugavet, A property of completely continuous operators in the space $C$, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 157–158 (Russian). MR 0157225
- Rainis Haller, Katriin Pirk, and Märt Põldvere, Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm, Acta Comment. Univ. Tartu. Math. 20 (2016), no. 1, 101–105. MR 3521706, DOI https://doi.org/10.12697/ACUTM.2016.20.08
- Rainis Haller, Johann Langemets, and Rihhard Nadel, Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums, Banach J. Math. Anal. 12 (2018), no. 1, 222–239. MR 3745582, DOI https://doi.org/10.1215/17358787-2017-0040
- Y. Ivakhno, V. Kadets Unconditional sums of spaces with bad projections, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 54 (2004), 30–35.
- V. Kadets, Banach spaces with the Daugavet property and Banach spaces with numerical index 1. Doctor of science thesis (in Russian). Kharkiv V. N. Karazin National University (2014), 307 pp. doi:10.13140/RG.2.1.2465.7689
- Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855–873. MR 1621757, DOI https://doi.org/10.1090/S0002-9947-99-02377-6
- Vladimir M. Kadets, Roman V. Shvidkoy, and Dirk Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), no. 3, 269–298. MR 1853772, DOI https://doi.org/10.4064/sm147-3-5
- Vladimir Kadets, Miguel Martín, Javier Merí, and Antonio Pérez, Spear operators between Banach spaces, Lecture Notes in Mathematics, vol. 2205, Springer, Cham, 2018. MR 3753623
- R. Nadel, Big slices of the unit ball in Banach spaces, Dissertationes Mathematicae Universitatis Tartuensis 132, University of Tartu Press, 2020, 109 pp.
- K. Pirk, Diametral diameter two properties, Daugavet-, and $\Delta$-points in Banach spaces, Dissertationes Mathematicae Universitatis Tartuensis 133, University of Tartu Press, 2020, 106 pp.
- R. V. Shvydkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), no. 2, 198–212. MR 1784413, DOI https://doi.org/10.1006/jfan.2000.3626
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2020): 46B20
Retrieve articles in all journals with MSC (2020): 46B20
Additional Information
Vladimir Kadets
Affiliation:
School of Mathematics and Informatics, V. N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine
MR Author ID:
202226
ORCID:
0000-0002-5606-2679
Email:
v.kateds@karazin.ua
Keywords:
Banach space,
the Daugavet property,
big slices,
convex combination of slices
Received by editor(s):
October 11, 2020
Published electronically:
March 22, 2021
Additional Notes:
The research was partially supported by project PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE), and on its final stage by National Research Foundation of Ukraine funded by Ukrainian State budget in frames of project 2020.02/0096 “Operators in infinite-dimensional spaces: the interplay between geometry, algebra and topology”.
Communicated by:
Stephen Dilworth
Article copyright:
© Copyright 2021
American Mathematical Society
