Functional Analysis
A class of Banach spaces with no unconditional basic sequence
[Une classe d'espace de Banach sans suite basique inconditionnelle]
Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 43-48.

Nous construisons un espace de Banach réflexif Xω1 ayant une base transfinie de longueur ω1 et n'admettant aucune suite basique inconditionnelle. De plus, tout opérateur borné d'un sous-espace de Xω1 dans cet espace est somme d'un opérateur diagonal très simple et d'un opérateur strictement singulier.

We give a construction of a reflexive Banach space Xω1 with a transfinite basis of length ω1 and with no unconditional basic sequence. In addition every bounded operator from a subspace of Xω1 into the space Xω1 is a sum of a simple diagonal operator and a strictly singular one.

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DOI : 10.1016/S1631-073X(03)00272-3
Argyros, Spiros A. 1 ; Lopez-Abad, Jordi 2 ; Todorcevic, Stevo 3

1 Department of Mathematics, National Technical University of Athens, Zogratou Campus, 15780 Athens, Greece
2 Équipe de logique mathématique, Université Paris VII, 2, place Jussieu, 75251 Paris cedex, France
3 CNRS–Université Paris VII, 2, place Jussieu, 75251 Paris cedex, France
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Argyros, Spiros A.; Lopez-Abad, Jordi; Todorcevic, Stevo. A class of Banach spaces with no unconditional basic sequence. Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 43-48. doi : 10.1016/S1631-073X(03)00272-3. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00272-3/

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