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Twisted sums with spaces
Authors:F Cabello Sá  nchez  J M F Castillo  N J Kalton  D T Yost
Institution:Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071 Badajoz, Spain ; Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071 Badajoz, Spain ; Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211 ; Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
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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