Tukey classification of some ideals on ω and the lattices of weakly compact sets in Banach spaces

We study the lattice structure of the family of weakly compact subsets of the unit ball BX of a separable Banach space X, equipped with the inclusion relation (this structure is denoted by K(BX)) and also with the parametrized family of “almost inclusion” relations K⊆L+εBX, where ε>0 (this structure is denoted by AK(BX)). Tukey equivalence between partially ordered sets and a suitable extension to deal with AK(BX) are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: K(BX) is equivalent either to a singleton, or to ωω, or to the family K(Q) of compact subsets of the rational numbers, or to the family [c]<ω of all finite subsets of the continuum. Also under the axiom of analytic determinacy, a similar classification of AK(BX) is obtained. For separable Banach spaces not containing ℓ1, we prove in ZFC that K(BX)∼AK(BX) are equivalent to either {0}, ωω, K(Q) or [c]<ω. The lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied.