Complemented ideals of $\ell_\infty$

TL;DR

This paper characterizes when certain subspaces of ll__ are complemented, linking this property to the approximability of associated Stone spaces and the separability of measure spaces.

Contribution

It provides a precise characterization of complemented ideals in ll__ using the concept of approximability of the related Stone space.

Findings

Identifies exactly which ideals yield complemented subspaces in ll__.

Connects complemented ideals to the separability of measure spaces on associated Stone spaces.

Provides a new criterion for the complemented property based on measure-theoretic and topological conditions.

Abstract

Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal I\subseteq \mathcal P(\omega)$ such that $c_{0,\mathcal I}$ is complemented in $\ell_\infty$ as exactly those ideals for which the space $K_{\mathcal I}= \mathsf{Stone}(\mathcal P(\omega)/\mathcal I)$ is approximable, i.e., the unit ball of the space $M(K_{\mathcal I})$ of signed Radon measures on $K_\mathcal I$ is separable in the weak* topology.