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

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 a countable family of maximal ideals, which we refer to as $\omega$-maximal ideals. We prove that if $c_{0,\mathcal{I}}$ admits a projection satisfying a certain condition, then $\mathcal{I}$ must be a special type of $\omega$-maximal ideal. Additionally, we characterize when the quotient space $c_{0,\mathcal{J}} / c_{0,\mathcal{I}}$ is finite-dimensional for two ideals $\mathcal{I} \subsetneq \mathcal{J}$.