A c₀-Saturated Banach Space with No Long Unconditional Basic Sequences

• Published in: Transactions of the American Mathematical Society 2009-09, Vol.361 (9), p.4541-4560
• Main Authors: Lopez-Abad, J., Todorcevic, S.
• Format: Article
• Language: English
• Subjects: Algebra; Banach space; Combinatorics; Exact sciences and technology; Functional analysis; Increasing sequences; Integers; Marital property; Mathematical analysis; Mathematical sequences; Mathematical vectors; Mathematics; Sciences and techniques of general use; Separable spaces
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-09-04858-2

We present a Banach space X with a Schauder basis of length X; i which is saturated by copies of co 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 w₁-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 w₁-distortable in the sense that for every λ > 1 there is an equivalent norm tex-math> on X such that for every non-separable subspace X of X there exist x,y ЄSx such that ≥ λ.