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 $C(K)$ spaces
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by F. Cabello Sánchez, J. M. F. Castillo, N. J. Kalton and D. T. Yost PDF

Trans. Amer. Math. Soc. 355 (2003), 4523-4541 Request permission

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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