m at h . FA ] 1 8 O ct 2 00 6 A c 0 - SATURATED BANACH SPACE WITH NO LONG UNCONDITIONAL BASIC SEQUENCES

We present a Banach space X with a Schauder basis of length ω1 which is saturated by copies of c0 and such that for every closed decomposition of a closed subspace X = X0 ⊕ X1, either X0 or X1 has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of X have " few operators " in the sense that every bounded operator T : X → X from a subspace X of X into X is the sum of a multiple of the inclusion and a ω1-singular operator, i.e., an operator S which is not an isomorphism on any non-separable subspace of X. We also show that while X is not distortable (being c0-saturated), it is arbitrarily ω1-distortable in the sense that for every λ > 1 there is an equivalent norm | · | on X such that for every non-separable subspace X of X there exist x, y ∈ SX such that |x|/|y| ≥ λ.