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 $\ell _\infty$.