Komlos Properties in Banach Lattices

TL;DR

This paper investigates Komlós-like properties in Banach lattices, showing specific failures of these properties in certain function spaces and constructing unbounded convex sets with particular convergence properties.

Contribution

It demonstrates the failure of the $oo$-pre-Komlós property in certain $C(K)$ spaces and constructs unbounded convex $uo$-pre-Komlós sets in infinite-dimensional Banach lattices.

Findings

$C(K)$ fails the $oo$-pre-Komlós property under certain topological conditions.

Existence of unbounded convex $uo$-pre-Komlós sets that are not $uo$-Komlós in infinite-dimensional Banach lattices.

Abstract

Several Koml\'os like properties in Banach lattices are investigated. We prove that $C(K)$ fails the $oo$-pre-Koml\'os property, assuming that the compact Hausdorff space $K$ has a nonempty separable open subset $U$ without isolated points such that every $u\in U$ has countable neighborhood base. We prove also that for any infinite dimensional Banach lattice $E$ there is an unbounded convex $uo$-pre-Koml\'os set $C\subseteq E_+$ which is not $uo$-Koml\'os.