On the existence of overcomplete sets in some classical nonseparable Banach spaces

Abstract

For a Banach space $X$ its subset $Y\subseteq X$ is called overcomplete if $|Y|=dens(X)$ and $Z$ is linearly dense in $X$ for every $Z\subseteq Y$ with $|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was introduced recently by T. Russo and J. Somaglia but overcomplete sets have been considered in separable Banach spaces since the 1950ties. We prove some absolute and consistency results concerning the existence and the nonexistence of overcomplete sets in some classical nonseparable Banach spaces. For example: $c_0(\omega_1)$, $C([0,\omega_1])$, $L_1(\{0,1\}^{\omega_1})$, $\ell_p(\omega_1)$, $L_p(\{0,1\}^{\omega_1})$ for $p\in (1, \infty)$ or in general WLD Banach spaces of density $\omega_1$ admit overcomplete sets (in ZFC). The spaces $\ell_\infty$, $\ell_\infty/c_0$, spaces of the form $C(K)$ for $K$ extremally disconnected, superspaces of $\ell_1(\omega_1)$ of density $\omega_1$ do not admit overcomplete sets (in ZFC). Whether the Johnson-Lindenstrauss space generatedin $\ell_\infty$ by $c_0$ and the characteristic functions of elements of an almost disjoint family of subsets of $\mathbb N$ of cardinality $\omega_1$ admits an overcomplete set is undecidable. The same refers to all nonseparable Banach spaces with the dual balls of density $\omega_1$ which are separable in the weak$^*$ topology. The results proved refer to wider classes of Banach spaces but several natural open questions remain open.