Grothendieck $C(K)$-spaces and the Josefson--Nissenzweig theorem

TL;DR

This paper introduces the $ ext{l}_1$-Grothendieck property for $C(K)$ spaces, characterizes it in terms of the Josefson--Nissenzweig theorem, and constructs examples illustrating its distinction from the classical Grothendieck property.

Contribution

It defines the $ ext{l}_1$-Grothendieck property, relates it to the Josefson--Nissenzweig theorem, and provides examples of spaces with this property that lack the classical Grothendieck property.

Findings

$C(K)$ has the $ ext{l}_1$-Grothendieck property iff no $ ext{l}_1$-sequence satisfies the Josefson--Nissenzweig conclusion.

Constructed a separable compact space $K$ where $C(K)$ has the $ ext{l}_1$-Grothendieck property but not the Grothendieck property.

Many Efimov spaces' $C(K)$ spaces do not have the $ ext{l}_1$-Grothendieck property.

Abstract

For a compact space $K$, the Banach space $C(K)$ is said to have the $\ell_1$-Grothendieck property if every weak* convergent sequence $\big\langle\mu_n\colon\ n\in\omega\big\rangle$ of functionals on $C(K)$ such that $\mu_n\in\ell_1(K)$ for every $n\in\omega$, is weakly convergent. Thus, the $\ell_1$-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that $C(K)$ has the $\ell_1$-Grothendieck property if and only if there does not exist any sequence of functionals $\big\langle\mu_n\colon\ n\in\omega\big\rangle$ on $C(K)$, with $\mu_n\in\ell_1(K)$ for every $n\in\omega$, satisfying the conclusion of the classical Josefson--Nissenzweig theorem. We construct an example of a separable compact space $K$ such that $C(K)$ has the $\ell_1$-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces $K$ their Banach spaces $C(K)$ do not have the $\ell_1$-Grothendieck property.