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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
DOI: https://doi.org/10.1090/S0002-9947-09-04858-2
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 $\| |\cdot \| |$ on $\mathfrak {X}$ such that for every non-separable subspace $X$ of $\mathfrak {X}$ there exist $x,y\in S_X$ such that $\| |x\| |/\| |y\| |\ge \lambda$.


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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
MR Author ID: 680200
Email: abad@logique.jussieu.fr

S. Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3
MR Author ID: 172980
Email: stevo@math.toronto.edu

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.