Topological characterization of Grothendieck spaces C(K)

Characterize, solely in terms of the topology of a compact space K, those compact spaces for which the Banach space C(K) has the Grothendieck property.

Background

The paper studies symmetric compactifications of the integers and shows that if a compact space K contains a copy of a symmetric compactification, then C(K) is not Grothendieck. The Grothendieck property for C(K) is a Banach-space property concerning weak* null sequences in C(K)^* and weak compactness of operators, whereas the stated problem asks for a characterization using only the topology of K.

The authors identify this as a long-standing problem posed by Diestel and place their results among known sufficient criteria and forbidden topological subspaces. Their results provide additional examples of compact spaces whose function spaces fail to be Grothendieck, but do not yield a complete topological characterization.