Complemented ideals of ${\ell }_{\infty }$

Abstract

Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal{I}\subseteq \mathcal{P}(\ddot{I}‰)$ such that $c_{0,\mathcal{I}}$ is complemented in ${\ell }_{\infty }$ as exactly those ideals for which the space $K_{\mathcal{I}}=\mathsf{Stone}(\mathcal{P}(\ddot{I}‰)/\mathcal{I})$ 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.