Functional Analysis
A class of Banach spaces with no unconditional basic sequence
[Une classe d'espace de Banach sans suite basique inconditionnelle]

Présenté par : Michel Talagrand

Argyros, Spiros A.1 ; Lopez-Abad, Jordi2 ; Todorcevic, Stevo3
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
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.

Reçu le :
Accepté le :
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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://www.numdam.org/articles/10.1016/S1631-073X(03)00272-3/

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

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

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

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

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

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

[10] Todorcevic, S. 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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