Grothendieck $C(K)$-spaces and the Josefson–Nissenzweig theorem
Volume 263 / 2023
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 $\langle \mu _n\colon n\in \omega \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 $\langle \mu _n\colon n\in \omega \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.
