Weak${}^{\ast }$-sequential properties of Johnson-Lindenstrauss spaces

Abstract

A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak${}^{\ast }$-closure of a convex bounded set $C\subseteq X^{\ast }$ is the weak${}^{\ast }$-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${}^{\ast }$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost.