A c_0-SATURATED BANACH SPACE WITH NO LONGUNCONDITIONAL BASIC SEQUENCES

摘要

We present a Banach space X with a Schauder basis of length ω_1which is saturated by copies of c_0 and such that for every closed decompositionof a closed subspace X = X_0(direct +)X_1, eitherX_0orX_1has to be separable. Thiscan be considered as the non-separable counterpart of the notion of hereditarilyindecomposable space. Indeed, the subspaces of X have "few operators" in thesense that every bounded operator T : X → X from a subspace X of X intoX is the sum of a multiple of the inclusion and a ω_1-singular operator, i.e.,an operator S which is not an isomorphism on any non-separable subspaceof X. We also show that while X is not distortable (being c_0-saturated),it is arbitrarily ω_1-distortable in the sense that for every λ > 1 there is anequivalent norm |||_|||on X such that for every non-separable subspaceXofX there exist x,y ∈S_Xsuch that |||x|||/|||y|||≥λ.