Twisted sums with 𝐶(𝐾) spaces

Sánchez, F.; Castillo, J.; Kalton, N.; Yost, D.

doi:10.1090/S0002-9947-03-03152-0

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
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Abstract: If 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 or $\omega^{\omega}.$ 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 $\ell_1$ . If $K=\omega^{\omega}$ 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 $C(\omega^{\omega})$ with strictly singular quotient map.

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