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

2014/06/20 by Antonio Avilés, Avilés, Antonio, Grzegorz Plebanek +3
Mathematics · #03E60 #03E75 #06A06 #46B20 (Primary) #46B50 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1406.5526

openalex publication_date 2014/06/20 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

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 K(B_X)) and also with the parametrized family of almost inclusion relations K ⊆ L+εB_X, where ε>0 (this structure is denoted by AK(B_X)). Tukey equivalence between partially ordered sets and a suitable extension to deal with AK(B_X) are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: K(B_X) is equivalent either to a singleton, or to ω^ω, or to the family K(ℚ) of compact subsets of the rational numbers, or to the family [\mathfrakc]^

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