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$F_σ$-ideals, colorings, and representation in Banach spaces

Jordi Lopez-Abad, Víctor Olmos-Prieto, Carlos Uzcátegui-Aylwin

TL;DR

The paper develops a comprehensive framework connecting $F_\sigma$-ideals and non-pathological ideals with representations by sequences in Banach spaces, notably through $\\mathcal{B}((x_n))$ and $\\mathcal{C}((x_n))$. It provides effective representations of these ideals in $C([0,1])$ and $C(2^{\mathbb{N}})$, highlighting the pivotal role of the space $c_0$ in determining when tall $\\mathcal{B}$-ideals can be realized. By introducing $c$-coloring ideals, the authors uncover a rich combinatorial structure linking Ramsey theory, graph/hypergraph colorings, and ideal pathology, proving that many tall $\\mathcal{B}$-ideals contain $c$-coloring ideals and that some tall ideals are not representable in $c_0$. They further show that tall $\\mathcal{B}$-ideals representable in $C(K)$ with $K$ countable necessarily carry a non-pathological front, yielding Borel selectors, and provide a cotype-based framework that constrains $\\mathcal{B}$-representations in spaces with nontrivial cotype. Overall, the work clarifies how the combinatorial properties of colorings interact with Banach-space representations to control the pathology and structure of these ideals.

Abstract

In recent works by L. Drewnowski and I. Labuda and J. Martínez et al., non-pathological analytic \( P \)-ideals and non-pathological \( F_σ\)-ideals have been characterized and studied in terms of their representations by a sequence \( (x_n)_n \) in a Banach space, as \( \mathcal{C}((x_n)_n) \) and \( \mathcal{B}((x_n)_n) \). The ideal \( \mathcal{C}((x_n)_n) \) consists of sets where the series \( \sum_{n \in A} x_n \) is unconditionally convergent, while \( \mathcal{B}((x_n)_n) \) involves weak unconditional convergence. In this paper, we further study these representations and provide effective descriptions of \( \mathcal{B} \)- and \( \mathcal{C} \)-ideals in the universal spaces \( C([0,1]) \) and \( C(2^{\mathbb{N}}) \), addressing a question posed by Borodulin-Nadzieja et al. A key aspect of our study is the role of the space \( c_0 \) in these representations. We focus particularly on \( \mathcal{B} \)-representations in spaces containing many copies of \( c_0 \), such as \( c_0 \)-saturated spaces of continuous functions. A central tool in our analysis is the concept of \( c \)-coloring ideals, which arise from homogeneous sets of continuous colorings. These ideals, generated by homogeneous sets of 2-colorings, exhibit a rich combinatorial structure. Among our results, we prove that for \( d \geq 3 \), the random \( d \)-homogeneous ideal is pathological, we construct hereditarily non-pathological universal \( c \)-coloring ideals, and we show that every \( \mathcal{B} \)-ideal represented in \( C(K) \), for \( K \) countable, contains a \( c \)-coloring ideal. Furthermore, by leveraging \( c \)-coloring ideals, we provide examples of \( \mathcal{B} \)-ideals that are not \( \mathcal{B} \)-representable in \( c_0 \). These findings highlight the interplay between combinatorial properties of ideals and their representations in Banach spaces.

$F_σ$-ideals, colorings, and representation in Banach spaces

TL;DR

The paper develops a comprehensive framework connecting -ideals and non-pathological ideals with representations by sequences in Banach spaces, notably through and . It provides effective representations of these ideals in and , highlighting the pivotal role of the space in determining when tall -ideals can be realized. By introducing -coloring ideals, the authors uncover a rich combinatorial structure linking Ramsey theory, graph/hypergraph colorings, and ideal pathology, proving that many tall -ideals contain -coloring ideals and that some tall ideals are not representable in . They further show that tall -ideals representable in with countable necessarily carry a non-pathological front, yielding Borel selectors, and provide a cotype-based framework that constrains -representations in spaces with nontrivial cotype. Overall, the work clarifies how the combinatorial properties of colorings interact with Banach-space representations to control the pathology and structure of these ideals.

Abstract

In recent works by L. Drewnowski and I. Labuda and J. Martínez et al., non-pathological analytic -ideals and non-pathological -ideals have been characterized and studied in terms of their representations by a sequence \( (x_n)_n \) in a Banach space, as \( \mathcal{C}((x_n)_n) \) and \( \mathcal{B}((x_n)_n) \). The ideal \( \mathcal{C}((x_n)_n) \) consists of sets where the series is unconditionally convergent, while \( \mathcal{B}((x_n)_n) \) involves weak unconditional convergence. In this paper, we further study these representations and provide effective descriptions of - and -ideals in the universal spaces \( C([0,1]) \) and \( C(2^{\mathbb{N}}) \), addressing a question posed by Borodulin-Nadzieja et al. A key aspect of our study is the role of the space in these representations. We focus particularly on -representations in spaces containing many copies of , such as -saturated spaces of continuous functions. A central tool in our analysis is the concept of -coloring ideals, which arise from homogeneous sets of continuous colorings. These ideals, generated by homogeneous sets of 2-colorings, exhibit a rich combinatorial structure. Among our results, we prove that for , the random -homogeneous ideal is pathological, we construct hereditarily non-pathological universal -coloring ideals, and we show that every -ideal represented in \( C(K) \), for countable, contains a -coloring ideal. Furthermore, by leveraging -coloring ideals, we provide examples of -ideals that are not -representable in . These findings highlight the interplay between combinatorial properties of ideals and their representations in Banach spaces.
Paper Structure (15 sections, 44 theorems, 128 equations)

This paper contains 15 sections, 44 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tallness and c0-saturation
  4. B-representability vs summability
  5. Representing non-pathological Fsigma ideals on spaces of continuous functions
  6. An explicit representation in C[0,1] and C2N
  7. Coloring ideals
  8. Examples of pathological c-coloring ideals
  9. Examples of non-pathological c-coloring ideals
  10. Universal d-colorings
  11. Tall colorings
  12. The block sequence coloring and c0
  13. A tall B-ideal not representable in c0
  14. Ideals B-representable in C(K), K countable
  15. An ideal characterization of c0

Key Result

Theorem 2.1

GrebikUzca2018 The tall family $\hom(c)$ of homogeneous sets of a coloring $c: [\mathbb{N}]^2\to \{0,1\}$ admits a Borel selector. ∎

Theorems & Definitions (107)

  • Theorem 2.1
  • Definition 2.2: Representability
  • Example 2.3
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • ...and 97 more