Comptes Rendus
Functional Analysis
Slicely countably determined Banach spaces
Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1277-1280.

A (separable) Banach space X is slicely countably determined if for every convex bounded subset A of X there is a sequence of slices (Sn) such that each slice of A contains one of the Sn. SCD-spaces form a joint generalization of spaces not containing 1 and those having the Radon–Nikodým property. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1.

Un espace de Banach (séparable) X est appelé un espace SCD si pour tout sous-ensemble A de X qui soit convexe et borné, il existe une suite Sn de tranches de A (une tranche est l'intersection non-vide de A avec un demi-espace ouvert de X) telle que chaque tranche de A contient une des tranches Sn. Les espaces SCD sont une généralisation des espaces qui ne contiennent pas 1 et aussi des espaces avec la propriété de Radon–Nikodým. On présente beacoup d'examples et diverses propriétés de cette classe. On donne aussi quelques applications aux espaces de Banach avec la propriété de Daugavet et la propriété alternative de Daugavet, aux espaces luxuriants (lush spaces), et aux espaces avec indice numérique 1.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.09.010

Antonio Avilés 1; Vladimir Kadets 2; Miguel Martín 3; Javier Merí 3; Varvara Shepelska 2

1 Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
2 Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
3 Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
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     title = {Slicely countably determined {Banach} spaces},
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Antonio Avilés; Vladimir Kadets; Miguel Martín; Javier Merí; Varvara Shepelska. Slicely countably determined Banach spaces. Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1277-1280. doi : 10.1016/j.crma.2009.09.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.010/

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[4] V.M. Kadets; R.V. Shvidkoy; G.G. Sirotkin; D. Werner Espaces de Banach ayant la propriété de Daugavet, C. R. Acad. Sci. Paris, Ser. I, Volume 325 (1997), pp. 1291-1994

[5] V.M. Kadets; R.V. Shvidkoy; G.G. Sirotkin; D. Werner Banach spaces with the Daugavet property, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 855-873

[6] V.M. Kadets; R.V. Shvidkoy; D. Werner Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math., Volume 147 (2001), pp. 269-298

[7] M. Martín; T. Oikhberg An alternative Daugavet property, J. Math. Anal. Appl., Volume 294 (2004), pp. 158-180

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