Comptes Rendus
Functional Analysis
A class of Banach spaces with no unconditional basic sequence
Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 43-48.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00272-3

Spiros A. Argyros 1; Jordi Lopez-Abad 2; Stevo Todorcevic 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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Spiros A. Argyros; Jordi Lopez-Abad; Stevo Todorcevic. A class of Banach spaces with no unconditional basic sequence. Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 43-48. doi : 10.1016/S1631-073X(03)00272-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00272-3/

[1] S.A. Argyros; I. Deliyanni Examples of asymptotic ℓ1 Banach spaces, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 973-995

[2] S.A. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math., to appear

[3] S.A. Argyros, A. Tolias, Indecomposability and unconditionality in duality, Preprint

[4] S. Bellenot; R. Haydon; E. Odell Quasi reflexive and tree spaces constructed in the spirit of R.C. James, Contemp. Math., Volume 85 (1989), pp. 19-43

[5] G.A. Edgar A long James space, Proc. Conf. on Measure Theory, Lecture Notes in Math., 794, 1980, pp. 31-37

[6] W.T. Gowers A solution to Banach's hyperplane problem, Bull. London Math. Soc., Volume 28 (1996), pp. 297-304

[7] W.T. Gowers; B. Maurey The unconditional basic sequence problem, J. Amer. Math. Soc., Volume 6 (1993), pp. 851-874

[8] J. Lindenstrauss On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 967-970

[9] Th. Schlumprecht An arbitrarily distortable Banach space, Israel J. Math., Volume 76 (1991), pp. 81-95

[10] S. Todorcevic Partitioning pairs of countable ordinals, Acta Math., Volume 159 (1987), pp. 261-294

[11] S. Todorcevic, Coherent sequences, in: Handbook of Set Theory, to appear

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