2025/01/26 by J. López-Abad, Lopez-Abad, Jordi, Víctor Olmos-Prieto +3
Computer Science · Mathematics · #05D10 #40A05 #46B15 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Primary 03E05 #Rings, Modules, and Algebras #Secondary 03E15
paper · pdf · doi:10.48550/arxiv.2501.15643
openalex publication_date 2025/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In recent works by L. Drewnowski and I. Labuda and J. Martínez et al., non-pathological analytic \( P \)-ideals and non-pathological \( Fσ\)-ideals have been characterized and studied in terms of their representations by a sequence \( (xn)n \) in a Banach space, as \( C((xn)n) \) and \( B((xn)n) \). The ideal \( C((xn)n) \) consists of sets where the series \( ∑n ∈ A xn \) is unconditionally convergent, while \( B((xn)n) \) involves weak unconditional convergence. In this paper, we further study these representations and provide effective descriptions of \( B \)- and \( C \)-ideals in the universal spaces \( C([0,1]) \) and \( C(2ℕ) \), addressing a question posed by Borodulin-Nadzieja et al. A key aspect of our study is the role of the space \( c0 \) in these representations. We focus particularly on \( B \)-representations in spaces containing many copies of \( c0 \), such as \( c0 \)-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 \( d ≥ 3 \), the random \( d \)-homogeneous ideal is pathological, we construct hereditarily non-pathological universal \( c \)-coloring ideals, and we show that every \( B \)-ideal represented in \( C(K) \), for \( K \) countable, contains a \( c \)-coloring ideal. Furthermore, by leveraging \( c \)-coloring ideals, we provide examples of \( B \)-ideals that are not \( B \)-representable in \( c0 \). These findings highlight the interplay between combinatorial properties of ideals and their representations in Banach spaces.