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

Abstract

For a compact space K, the Banach space C(K) is said to have the ℓ1-Grothendieck property if every weak* convergent sequence µn : n ∈ ω of functionals on C(K) such that µn ∈ ℓ1(K) for every n ∈ ω, is weakly convergent. Thus, the ℓ1-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that C(K) has the ℓ1-Grothendieck property if and only if there does not exist any sequence of functionals µn : n ∈ ω on C(K), with µn ∈ ℓ1(K) for every n ∈ ω, 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 ℓ1-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 ℓ1-Grothendieck property.