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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A $ c_0$-saturated Banach space with no long unconditional basic sequences


Authors: J. Lopez-Abad and S. Todorcevic
Journal: Trans. Amer. Math. Soc. 361 (2009), 4541-4560
MSC (2000): Primary 46B20, 03E02; Secondary 46B26, 46B28
Published electronically: April 14, 2009
MathSciNet review: 2506418
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Abstract: We present a Banach space $ \mathfrak{X}$ with a Schauder basis of length $ \omega_1$ which is saturated by copies of $ c_0$ and such that for every closed decomposition of a closed subspace $ X=X_0\oplus X_1$, either $ X_0$ or $ X_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $ \mathfrak{X}$ have ``few operators'' in the sense that every bounded operator $ T:X \rightarrow \mathfrak{X}$ from a subspace $ X$ of $ \mathfrak{X}$ into $ \mathfrak{X}$ is the sum of a multiple of the inclusion and a $ \omega_1$-singular operator, i.e., an operator $ S$ which is not an isomorphism on any non-separable subspace of $ X$. We also show that while $ \mathfrak{X}$ is not distortable (being $ c_0$-saturated), it is arbitrarily $ \omega_{1}$-distortable in the sense that for every $ \lambda>1$ there is an equivalent norm $ \Vert \vert\cdot\Vert \vert$ on $ \mathfrak{X}$ such that for every non-separable subspace $ X$ of $ \mathfrak{X}$ there exist $ x,y\in S_X$ such that $ \Vert \vert x\Vert \vert/\Vert \vert y\Vert \vert\ge \lambda$.


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Additional Information

J. Lopez-Abad
Affiliation: Université Paris Diderot Paris 7, UFR de mathématiques case 7012, site Chevaleret, 75205 Paris Cedex 13, France
Address at time of publication: Instituto de Ciencias Matematicas, CSIC-UAM-UC3M-UCM, Consejo Superior de Investigationes Cientificas, c/Serrano 121, 28006, Madrid, Spain
Email: abad@logique.jussieu.fr

S. Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3
Email: stevo@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04858-2
PII: S 0002-9947(09)04858-2
Received by editor(s): January 26, 2007
Published electronically: April 14, 2009
Additional Notes: This work was supported by NSERC and CNRS.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.