Twisted sums with 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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: If is a separable Banach space, we consider the existence of non-trivial twisted sums
, where
or
For the case
we show that there exists a twisted sum whose quotient map is strictly singular if and only if
contains no copy of
. If
we prove an analogue of a theorem of Johnson and Zippin (for
) by showing that all such twisted sums are trivial if
is the dual of a space with summable Szlenk index (e.g.,
could be Tsirelson's space); a converse is established under the assumption that
has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with
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