On the complementation of spaces of $\mathcal I$-null sequences

Abstract

We study the complementation (in $\ell_\infty$) of the Banach space $c_{0,\mathcal{I}}$, consisting of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the supremum norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that the complementation of these spaces is related to a condition requiring that the ideal is the intersection of at most a countable family of maximal ideals, which we refer to as at most $\omega$-maximal ideals. We prove that $\mathcal{I}$ is at most $\omega$-maximal exactly when $c_{0,\mathcal{I}}$ is the kernel of an operator from $\ell_\infty$ to itself satisfying a certain property. In addition, we show that the existence of a Banach lattice isomorphism from the quotient $\ell_\infty/c_{0,\mathcal{I}}$ onto a closed sublattice of $\ell_\infty$ is equivalent to $\mathcal{I}$ being an at most $\omega$-maximal ideal. Moreover, $\ell_\infty/c_{0,\mathcal I}$ is Banach lattice isomorphic to $\ell_\infty$ if and only if $\mathcal I$ is the intersection of a certain countably infinite family of maximal ideals; we refer to such ideals as strongly $\omega$-maximal. Finally, for two ideals $\mathcal I\subsetneq\mathcal J$, we characterize when the quotient space $c_{0,\mathcal{J}} / c_{0,\mathcal{I}}$ is finite-dimensional.