Double commutants of multiplication operators on C(K).

الأصل · EN

Let C(K) be the space of all real or complex valued continuous functions on a compact Hausdorff space K. We are interested in the following property of K: for any real valued f ∈ C(K) the double commutant of the corresponding multiplication operator F coincides with the norm closed algebra generated by F and I. In this case we say that K ∈ DCP. It was proved in Ki that any locally connected metrizable continuum is in DCP. In this paper we indicate a class of arc connected but not locally connected continua that are in DCP. We also construct an example of a continuum that is not arc connected but is in DCP.