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 (ℓ ∞ → ℓ ∞ /c 0 ).We show through an application of Kalton's Monotone Transfinite Sequence Theorem that a quotient map of a subspace of ℓ ∞ of sequences that converge to zero along an ideal in 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 N, with one much larger than the other, occur in abundance.Some classical examples of ideals in N presented explicitly are: the finite subsets of N, the subsets of N with convergent reciprocal series, and, the subsets of N with density zero, Banach density zero or Buck density zero.

A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices

“…Structural properties of the set of bounded I-convergent sequences c(I) ∩ ℓ ∞ and its subspace c 0 (I) ∩ ℓ ∞ have been recently studied in the literature, sometimes providing answers to longstanding questions, see e.g. [5,14,18,22].…”