Nonatomic submeasures on $\mathbb N$ and the Banach space $\ell_1$

TL;DR

This paper introduces nonatomic submeasures derived from bounded sequences in , explores their structure as a closed subspace within , and examines related operator spaces in .

Contribution

It characterizes the space of nonatomic submeasures on  as a closed subspace of  and investigates associated operator spaces.

Findings

Nonatomic submeasures form a closed subspace of .

The space of relevant operators in  is analyzed.

New structural insights into  and submeasures are provided.

Abstract

Nonatomic bounded sequences in $\ell_1$, that is, those giving rise to nonatomic submeasures on $\mathbb N$ are introduced and shown to form a closed subspace nonat$(\ell_1)$ of $\ell_\infty(\ell_1)$, and some spaces of relevant operators in $\ell_1$ are considered.