A letter concerning Leonetti's paper `Continuous Projections onto Ideal Convergent Sequences'

Abstract: Leonetti proved that whenever $\mathcal I$ is an ideal on $\mathbb N$ such that there exists an~uncountable family of sets that are not in $\mathcal I$ with the property that the intersection of any two distinct members of that family is in $\mathcal I$, then the space $c_{0,\mathcal I}$ of sequences in $\ell_\infty$ that converge to 0 along $\mathcal I$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $\ell_\infty / c_{0,\mathcal I}$ does not even embed into $\ell_\infty$.