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 ${\ell }_{\infty }$.