Banach spaces of $\mathcal I$-convergent sequences

TL;DR

This paper investigates Banach spaces formed by sequences that are $ extit{I}$-convergent to zero, exploring their isometric properties, lattice structures, and connections to classical spaces and ideals, revealing new structural insights.

Contribution

It establishes isometry criteria for $c_{0, ext{I}}$ spaces based on ideal isomorphisms and links the Katětov pre-order to Banach lattice isometries, also characterizing all closed ideals of $ell_.

Findings

$c_{0, ext{I}}$ and $c_{0, ext{J}}$ are isometric iff $ ext{I}$ and $ ext{J}$ are isomorphic.

The Katětov pre-order $ ext{I} \leq_K ext{J}$ corresponds to Banach lattice isometries between $c_{0, ext{I}}$ and $c_{0, ext{J}}$.

Every closed ideal of $ell_$ is of the form $c_{0, ext{I}}$ for some ideal $ ext{I}$.

Abstract

We study the space $c_{0,\mathcal{I}}$ of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the sup norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that two such spaces, $c_{0,\mathcal{I}}$ and $c_{0,\mathcal{J}}$, are isometric exactly when the ideals $\mathcal{I}$ and $\mathcal{J}$ are isomorphic. Additionally, we analyze the connection of the well-known Kat\v{e}tov pre-order $\leq_K$ on ideals with some properties of the space $c_{0,\mathcal{I}}$. For instance, we show that $\mathcal{I}\leq_K\mathcal{J}$ exactly when there is a (not necessarily onto) Banach lattice isometry from $c_{0,\mathcal{I}}$ to $c_{0,\mathcal{J}}$, satisfying some additional conditions. We present some lattice-theoretic properties of $c_{0,\mathcal{I}}$, particularly demonstrating that every closed ideal of $\ell_\infty$ is equal to $c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$ on $\mathbb{N}$. We also show that certain classical Banach spaces are isometric to $c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$, such as the spaces $\ell_\infty(c_0)$ and $c_0(\ell_\infty)$. Finally, we provide several examples of ideals for which $c_{0,\mathcal{I}}$ is not a Grothendieck space.