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Delta- and Daugavet points in Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  27 February 2020

T. A. Abrahamsen ,
R. Haller ,
V. Lima  and
K. Pirk
T. A. Abrahamsen
Affiliation:
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway (trond.a.abrahamsen@uia.no)
R. Haller
Affiliation:
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia (rainis.haller@ut.ee; katriinp@ut.ee)
V. Lima
Affiliation:
Department of Engineering Sciences, University of Agder, Postboks 422, 4604 Kristiansand, Norway (Vegard.Lima@uia.no)
K. Pirk
Affiliation:
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia (rainis.haller@ut.ee; katriinp@ut.ee)
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Abstract

A Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.

Keywords

diametral diameter-two propertyDaugavet propertyL1-spaceL1-predual spaceMüntz space

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B04: Isometric theory of Banach spaces 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 475 - 496
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

1.Abrahamsen, T. A., Hájek, P., Nygaard, O., Talponen, J. and Troyanski, S., Diameter 2 properties and convexity, Studia Math. 232(3) (2016), 227–242.Google Scholar
2.Abrahamsen, T. A., Leerand, A., Martiny, A. and Nygaard, O., Two properties of Müntz spaces, Demonstr. Math. 50 (2017), 239–244.CrossRefGoogle Scholar
3.Abrahamsen, T. A., Lima, V. and Nygaard, O., Almost isometric ideals in Banach spaces, Glasgow Math. J. 56(2) (2014), 395–407.CrossRefGoogle Scholar
4.Abrahamsen, T. A., Lima, V., Nygaard, O. and Troyanski, S., Diameter two properties, convexity and smoothness, Milan J. Math. 84(2) (2016), 231–242.CrossRefGoogle Scholar
5.Becerra Guerrero, J., López-Pérez, G. and Rueda Zoca, A., Diametral diameter two properties in Banach spaces, J. Convex Anal. 25(3) (2018), 817–840.Google Scholar
6.Becerra Guerrero, J. and Martín, M., The Daugavet property of C*-algebras, JB*-triples, and of their isometric preduals, J. Funct. Anal. 224(2) (2005), 316–337.CrossRefGoogle Scholar
7.Becerra Guerrero, J. and Martín, M., The Daugavet property for Lindenstrauss spaces, in Methods in Banach space theory, London Mathematical Society Lecture Note Series, Volume 337, 91–96 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
8.Bilik, D., Kadets, V., Shvidkoy, R. and Werner, D., Narrow operators and the Daugavet property for ultraproducts, Positivity 9(1) (2005), 45–62.CrossRefGoogle Scholar
9.Borwein, P. and Erdélyi, T., Generalizations of Müntz's theorem via a Remez-type inequality for Müntz spaces, J. Amer. Math. Soc. 10(2) (1997), 327–349.CrossRefGoogle Scholar
10.Erdélyi, T., The ‘full Clarkson–Erdös–Schwartz theorem’ on the closure of non-dense Müntz spaces, Studia Math. 155(2) (2003), 145–152.CrossRefGoogle Scholar
11.Haller, R., Langemets, J. and Nadel, R., Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums, Banach J. Mat. Anal. 12(1) (2018), 222–239.CrossRefGoogle Scholar
12.Ivakhno, Y. and Kadets, V. M., Unconditional sums of spaces with bad projections, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 645(54) (2004), 30–35.Google Scholar
13.Jameson, G. J. O., Counting zeros of generalised polynomials, Math. Gazette 90 (2006), 223–234.CrossRefGoogle Scholar
14.Kadets, V. M., Shvidkoy, R. V., Sirotkin, G. G. and Werner, D., Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352(2) (2000), 855–873.CrossRefGoogle Scholar
15.Lindenstrauss, J., Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964).Google Scholar
16.Martiny, A., On octahedraliy and Müntz spaces, Math. Scand. to appear, arXiv e-prints (2018).Google Scholar
17.Werner, D., Recent progress on the Daugavet property, Irish Math. Soc. Bull. 46 (2001), 77–97.Google Scholar
18.Werner, D., A remark about Müntz spaces, Preprint (2008).Google Scholar