Twisted Sums with C(k) Spaces

If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 → C(K) → Y → X → 0, where K = [0, 1] or ωω . 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 `1. If K = ωω 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(ωω) with strictly singular quotient map.