Twisted sums with 𝐶(𝐾) spaces

Abstract

If X X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 → C ( K ) → Y → X → 0 0\to C(K)\to Y\to X\to 0 , where K = [ 0 , 1 ] K=[0,1] or ω ω . \omega ^{\omega }. For the case K = [ 0 , 1 ] K=[0,1] we show that there exists a twisted sum whose quotient map is strictly singular if and only if X X contains no copy of ℓ 1 \ell _1 . If K = ω ω K=\omega ^{\omega } we prove an analogue of a theorem of Johnson and Zippin (for K = [ 0 , 1 ] K=[0,1] ) by showing that all such twisted sums are trivial if X X is the dual of a space with summable Szlenk index (e.g., X X could be Tsirelson’s space); a converse is established under the assumption that X X has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with C ( ω ω ) C(\omega ^{\omega }) with strictly singular quotient map.