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 $\mathcal{I}⊊\mathcal{J}$, we characterize when the quotient space $c_{0,\mathcal{J}}/c_{0,\mathcal{I}}$ is finite-dimensional.