Complemented ideals of $\ell_\infty$

TL;DR

The paper studies when the closed ideal $c_{0,I}$ of $\ell_\infty$ is complemented, translating the problem into the topology of the Boolean Stone space $K_I=Stone(P(omega)/I)$. It proves a suite of equivalent conditions: $c_{0,I}$ complemented ⇔ $K_I$ approximable ⇔ $C(K_I)$ embeds into $\ell_\infty$ (isometric or isomorphic), together with a combinatorial covering property of $(P(omega)/I)^+$. The authors connect Boolean-algebra approximability, weak* separability of $M(K_I)$, and Banach-space embeddability, and provide examples showing the sharpness of the equivalences and the limits of the approach. These results yield new insights into Grothendieck properties and the structure of complemented subspaces within $\ell_\infty$, and raise further questions about measures of countable Maharam type in this setting.

Abstract

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

Theorems & Definitions (17)

• Theorem 1
• Theorem 1.1: Kelley Kelley
• Theorem 1.3: Goodner-Nachbin, see Kalton
• Proposition 1.4: Kalton
• Theorem 1.5: Talagrand
• Theorem 1.6: Lindenstrauss, see Kalton
• Proposition 2.1
• proof
• Proposition 2.2
• proof