Daugavet- and delta-points in Banach spaces with unconditional bases
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- by Trond A. Abrahamsen, Vegard Lima, André Martiny and Stanimir Troyanski HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 379-398
Abstract:
We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a $1$-unconditional basis. A norm one element $x$ in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance $2$ from $x$. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than $2$.
We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a $1$-unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a $1$-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.
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Additional Information
- Trond A. Abrahamsen
- Affiliation: Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway
- MR Author ID: 773387
- ORCID: 0000-0003-1010-0040
- Email: trond.a.abrahamsen@uia.no
- Vegard Lima
- Affiliation: Department of Engineering Sciences, University of Agder, Postboks 509, 4898 Grimstad, Norway
- MR Author ID: 723061
- Email: Vegard.Lima@uia.no
- André Martiny
- Affiliation: Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway
- Email: andre.martiny@uia.no
- Stanimir Troyanski
- Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Science, bl.8, acad. G. Bonchev str. 1113 Sofia, Bulgaria; and Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo (Murcia), Spain
- MR Author ID: 174580
- Email: stroya@um.es
- Received by editor(s): July 11, 2020
- Received by editor(s) in revised form: January 18, 2021
- Published electronically: April 28, 2021
- Additional Notes: The fourth-named author was supported by MTM2017-86182-P (AEI/FEDER, UE), and Bulgarian National Scientific Fund, Grant, KP–06–H22/4, 04.12.2018.
- © Copyright 2021 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 379-398
- MSC (2020): Primary 46B20, 46B22, 46B04
- DOI: https://doi.org/10.1090/btran/68
- MathSciNet review: 4249632