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

Antonio Avilés, Grzegorz Plebanek+1 more|arXiv (Cornell University)|Jun 20, 2014

Advanced Banach Space Theory25 references3 citations

TL;DR

This paper classifies the lattice structures of weakly compact subsets in separable Banach spaces using Tukey reducibility under the axiom of analytic determinacy. It establishes that for such spaces, the structure K(BX) is Tukey equivalent to one of four canonical posets: a singleton, ωω, K(Q), or [c]<ω, with a refined classification for AK(BX) involving metric-based almost-inclusion relations. The key contribution is a complete Tukey classification under analytic determinacy, with ZFC results for ℓ₁-free 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.

Motivation & Objective

• Classify the lattice structure of weakly compact subsets in separable Banach spaces using Tukey reducibility.
• Extend the classification to include metric structure via the AK(BX) framework, which incorporates ε-neighborhoods of sets.
• Establish that under the axiom of analytic determinacy, K(BX) is Tukey equivalent to one of four canonical posets: {0}, ωω, K(Q), or [c]<ω.
• Provide a finer classification for AK(BX), distinguishing cases where the structure reflects stronger geometric or topological properties.
• Identify conditions under which K(BX) and AK(BX) are Tukey equivalent, particularly in spaces not containing ℓ₁.
• Address the consistency of the classification in the absence of analytic determinacy, and explore open problems in this context.

Proposed method

• Use Tukey reducibility to compare partially ordered sets representing weakly compact families in Banach spaces.
• Introduce AK(BX), a structure equipped with parametrized relations K ⊆ L + εBX for ε > 0, to incorporate metric information.
• Apply a Lusin gap dichotomy for coanalytic sets to analyze complex ideals on ω, leading to the classification of I⊥ for analytic families I.
• Utilize the axiom of analytic determinacy to ensure the existence of certain homogeneous substructures, enabling the classification into canonical types.
• Apply Ramsey-theoretic arguments and tree analysis (e.g., antichain and chain decompositions in 2<ω) to control the structure of families of sets.
• Construct a counterexample in a model with MAℵ₁ and Lusin’s hypothesis to show that [ω₁]<ω is not Tukey equivalent to the canonical posets under certain cardinal invariants.

Experimental results

Research questions

• RQ1 Under the axiom of analytic determinacy, what are the possible Tukey types of the lattice K(BX) for separable Banach spaces?
• RQ2 How does the introduction of metric-based relations in AK(BX) refine the classification of weakly compact sets compared to K(BX)?
• RQ3 Is it consistent that there exists a non-reflexive separable Banach space for which K(BX) is not Tukey equivalent to any of {0}, ωω, K(Q), or [c]<ω?
• RQ4 Can the equivalence between K(BX) and AK(BX) be established in ZFC for Banach spaces not containing ℓ₁?
• RQ5 Does the structure AK(BX) always satisfy ωω ⪯ AK(BX) or AK(BX) ⪯ ω in non-SWCG spaces, or are there intermediate types?

Key findings

• Under the axiom of analytic determinacy, the lattice K(BX) of weakly compact subsets of the unit ball BX is Tukey equivalent to one of four canonical posets: a singleton (corresponding to reflexive spaces), ωω, K(Q), or [c]<ω.
• AK(BX), which incorporates metric-based almost-inclusion relations, admits a refined classification into five types: {0}, ω, ωω, K(Q), and [c]<ω, with the case ω corresponding to strongly weakly compactly generated spaces.
• For separable Banach spaces not containing ℓ₁, the structures K(BX) and AK(BX) are Tukey equivalent in ZFC, and each is equivalent to one of {0}, ωω, K(Q), or [c]<ω.
• In a model satisfying MAℵ₁ and Lusin’s hypothesis, the poset [ω₁]<ω is not Tukey equivalent to any of {0}, ω, ωω, K(Q), or [c]<ω, due to differing cofinalities (ℵ₁ vs. ℵ₀, d, or c).
• The construction in Theorem 7.12 yields a non-reflexive, separable Banach space with an unconditional basis whose weakly compact sets yield a structure not equivalent to any of the four canonical types, assuming the existence of a coanalytic set of size ℵ₁.
• Under the same model, AK(BX) for X = ℓ₁(ω₁) has cofinality ω₁, and since cf(ωω) = d, it follows that ωω ⪯ AK(BX) fails if ω₁ < d, showing that AK(BX) can escape the canonical classes.