Weak$^*$-sequential properties of Johnson-Lindenstrauss spaces

TL;DR

This paper investigates the weak$^*$-sequential properties of Johnson-Lindenstrauss spaces, showing under the Continuum Hypothesis that these spaces can either have or lack Efremov's property ($ ext{E}$), thus addressing open questions in Banach space theory.

Contribution

It demonstrates, assuming the Continuum Hypothesis, the existence of Johnson-Lindenstrauss spaces with and without Efremov's property ($ ext{E}$), resolving previous gaps and questions.

Findings

Existence of Johnson-Lindenstrauss spaces with property ($ ext{E}$) under CH.

Existence of Johnson-Lindenstrauss spaces without property ($ ext{E}$) under CH.

Addresses questions posed by Plichko and Yost.

Abstract

A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property ($\mathcal{E}$). This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost.