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 $\mathcal{AK}(B_X)$). Tukey equivalence between partially ordered sets and a suitable extension to deal with $\mathcal{AK}(B_X)$ 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 $\mathcal{AK}(B_X)$ is obtained. For separable Banach spaces not containing $\ell^1$, we prove in ZFC that $\mathcal{K}(B_X) \sim \mathcal{AK}(B_X)$ 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.