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

For a compact space $K$, the Banach space $C(K)$ is said to have the ${\mathrm{\ell }}_1$-Grothendieck property if every weak* convergent sequence $⟨{\mu }_n\text{\hspace{0.17em}⁣}:n\in \omega ⟩$ of functionals on $C(K)$ such that ${\mu }_n\in {\mathrm{\ell }}_1(K)$ for every $n\in \omega$, is weakly convergent. Thus, the ${\mathrm{\ell }}_1$-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that $C(K)$ has the ${\mathrm{\ell }}_1$-Grothendieck property if and only if there does not exist any sequence of functionals $⟨{\mu }_n\text{\hspace{0.17em}⁣}:n\in \omega ⟩$ on $C(K)$, with ${\mu }_n\in {\mathrm{\ell }}_1(K)$ 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 ${\mathrm{\ell }}_1$-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 ${\mathrm{\ell }}_1$-Grothendieck property.