A 𝑐₀-saturated Banach space with no long unconditional basic sequences

Abstract

We present a Banach space X \mathfrak X with a Schauder basis of length ω 1 \omega _1 which is saturated by copies of c 0 c_0 and such that for every closed decomposition of a closed subspace X = X 0 ⊕ X 1 X=X_0\oplus X_1 , either X 0 X_0 or X 1 X_1 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 \mathfrak X have “few operators” in the sense that every bounded operator T : X → X T:X \rightarrow \mathfrak {X} from a subspace X X of X \mathfrak {X} into X \mathfrak {X} is the sum of a multiple of the inclusion and a ω 1 \omega _1 -singular operator, i.e., an operator S S which is not an isomorphism on any non-separable subspace of X X . We also show that while X \mathfrak {X} is not distortable (being c 0 c_0 -saturated), it is arbitrarily ω 1 \omega _{1} -distortable in the sense that for every λ > 1 \lambda >1 there is an equivalent norm ‖ | ⋅ ‖ | \| |\cdot \| | on X \mathfrak {X} such that for every non-separable subspace X X of X \mathfrak {X} there exist x , y ∈ S X x,y\in S_X such that ‖ | x ‖ | / ‖ | y ‖ | ≥ λ \| |x\| |/\| |y\| |\ge \lambda .