Extremely non-nice operators into \(CK\) and continuous mappings into the Hilbert cube (Q793317)

We present a topological characterization of those compact spaces \(K\) that admit an extremely non-nice compact operator into \(CK\), i.e. an extremal compact operator \(T\) from some Banach space \(X\) into \(CK\) whose composition with no extremal functional on \(CK\) is an extremal functional on \(X\). This topological property, namely the existence of a continuous mapping with nowhere dense fibers into the Hilbert cube, is related to the corresponding property with the Hilbert cube replaced by the unit interval. In this way we show that some large classes of compact spaces admit extremely non-nice operators, e.g. the separable perfect and the hyperstonean perfect spaces. Some spaces without this property are presented.