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Twisted sums with $C(K)$ spaces


Authors: F. Cabello Sánchez, J. M. F. Castillo, N. J. Kalton and D. T. Yost
Journal: Trans. Amer. Math. Soc. 355 (2003), 4523-4541
MSC (2000): Primary 46B03, 46B20
DOI: https://doi.org/10.1090/S0002-9947-03-03152-0
Published electronically: July 2, 2003
MathSciNet review: 1990760
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Abstract: If $X$ is a separable Banach space, we consider the existence of non-trivial twisted sums $0\to C(K)\to Y\to X\to 0$, where $K=[0,1]$ or $\omega^{\omega}.$For the case $K=[0,1]$ we show that there exists a twisted sum whose quotient map is strictly singular if and only if $X$ contains no copy of $\ell_1$. If $K=\omega^{\omega}$ we prove an analogue of a theorem of Johnson and Zippin (for $K=[0,1]$) by showing that all such twisted sums are trivial if $X$ is the dual of a space with summable Szlenk index (e.g., $X$ could be Tsirelson's space); a converse is established under the assumption that $X$ has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with $C(\omega^{\omega})$ with strictly singular quotient map.


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Additional Information

F. Cabello Sánchez
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071 Badajoz, Spain
Email: fcabello@unex.es

J. M. F. Castillo
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071 Badajoz, Spain
Email: castillo@unex.es

N. J. Kalton
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: nigel@math.missouri.edu

D. T. Yost
Affiliation: Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Email: dthoyost@ksu.edu.sa

DOI: https://doi.org/10.1090/S0002-9947-03-03152-0
Received by editor(s): June 21, 2001
Received by editor(s) in revised form: June 5, 2002
Published electronically: July 2, 2003
Additional Notes: The research of the first two authors was supported in part by the DGICYT project BFM 2001-0387
The third author was supported by NSF grant DMS-9870027.
The fourth author was supported substantially by the Junta de Extremadura, and for a few days by Research Centre Project Number Math/1420/25 from his present institution
Article copyright: © Copyright 2003 American Mathematical Society

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