Saturated extensions, the attractors method and Hereditarily James Tree Spaces

Abstract

Contents 0. Introduction 1 0.1. Hereditarily James Tree spaces 3 0.2. Saturated extensions 7 0.3. The attractors method 10 1. Strictly singular extensions with attractors 15 2. Strongly strictly singular extensions 24 3. The James tree space JT F2 . 32 4. The space (X F2 ) * and the space of the operators L((X F2 ) * ) 40 5. The structure of X * F2 and a variant of X F2 46 6. A nonseparable HI space with no reflexive subspace 52 7. A HJT space with unconditionally and reflexively saturated dual 63 Appendix A. The auxiliary space and the basic inequality 68 Appendix B. The James tree spaces JT F2,s , JT F2 and JT Fs 75 References 86 Research partially supported by EPEAEK program "PYTHAGORAS".

On spaces admitting no ℓ p or c 0 spreading model

“…For an arbitrary seminormalized weakly null sequence in Z X we can pass to a subsequence satisfying the same (1), (2) and (3) as in the proof of Proposition 7. Since Remark 2 is the same are Remark 1 with a different quotient map, by mimicking the proof of Proposition 7 it can be shown that this subsequence in Cesaro summable, as required.…”

“…In [2] a Banach space Z is constructed satisfying the assumptions of Theorem 21. Z * is the desired space.…”

“…Namely, we observe that a space constructed in [2] does not admit as quotient any space with separable dual. This solves a question posed in [13, page 86, Remark IV.1].…”

“…(2) There is a separable space Y not admitting any ℓ p for 1 ≤ p < ∞ or c 0 as a spreading model such that X is a quotient of Y .…”

A Banach space dichotomy theorem for quotients of subspaces

“…Indeed, if X is QHI restricted to block-subspaces, then no quotient of X by an FDD-block subspace can contain an unconditional basic sequence, and therefore X must contain a quotient of a subspace with the stronger "angle zero" property. [1], the distinction between general quotient spaces and quotients by FDD-block subspaces may be essential: there exists a separable dual space X with a Schauder basis such that quotients with w * -closed kernels are HI, yet every quotient has a further quotient isomorphic to l 2 . Since FDD-block subspaces of X are w * -closed, this space has the restricted QHI property, but it is not QHI by the l 2 -saturation property.…”