Tukey classification of some ideals in $\omega$ and the lattices of weakly compact sets in Banach spaces

Abstract

We study the lattice structure of the family of weakly compact subsets of the unit ball $B_X$ of a separable Banach space $X$, equipped with the inclusion relation (this structure is denoted by $\mathcal{K}(B_X)$) and also with the parametrized family of almost inclusion relations $K\subseteq L+\epsilon B_X$, where $\epsilon >0$ (this structure is denoted by $\mathcal{A}\mathcal{K}(B_X)$). Tukey equivalence between partially ordered sets and a suitable extension to deal with $\mathcal{A}\mathcal{K}(B_X)$ are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: $\mathcal{K}(B_X)$ is equivalent either to a singleton, or to ${\omega }^{\omega }$, or to the family $\mathcal{K}(\mathbb{Q})$ of compact subsets of the rational numbers, or to the family $[\mathfrak{c}{]}^{<\omega }$ of all finite subsets of the continuum. Also under the axiom of analytic determinacy, a similar classification of $\mathcal{A}\mathcal{K}(B_X)$ is obtained. For separable Banach spaces not containing ${\ell }^1$, we prove in ZFC that $\mathcal{K}(B_X)\sim \mathcal{A}\mathcal{K}(B_X)$ are equivalent to either $\{0\}$, ${\omega }^{\omega }$, $\mathcal{K}(\mathbb{Q})$ or $[\mathfrak{c}{]}^{<\omega }$. The lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied.

arXiv commentThe main open problem stated in the first version of the preprint is now solved