A family of quotient maps of ${\ell }^{\infty }$ that do not admit uniformly continuous right inverses

Abstract

Previously only two examples of Banach space quotient maps which do not admit uniformly continuous right inverses were known: one due to Aharoni and Lindenstrauss and one due to Kalton (${\ell }^{\infty }\to {\ell }^{\infty }/c_0$). We show through an application of Kalton's Monotone Transfinite Sequence Theorem that a quotient map of a subspace of ${\ell }^{\infty }$ of sequences that converge to zero along an ideal in $\mathbb{N}$ toward another such subspace, provided one of the ideals is `much larger' than the other, cannot have a uniformly continuous right inverse. We show in general that pairs of ideals in $\mathbb{N}$, with one much larger than the other, occur in abundance. Some classical examples of ideals in $\mathbb{N}$ presented explicitly are: the finite subsets of $\mathbb{N}$, the subsets of $\mathbb{N}$ with convergent reciprocal series, and, the subsets of $\mathbb{N}$ with density zero, Banach density zero or Buck density zero.