Abstract
If X X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 â C ( K ) â Y â X â 0 0\to C(K)\to Y\to X\to 0 , where K = [ 0 , 1 ] K=[0,1] or Ï Ï . \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 â 1 \ell _1 . If K = Ï Ï 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 Tsirelsonâs 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 ( Ï Ï ) C(\omega ^{\omega }) with strictly singular quotient map.
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