Double commutants of multiplication operators on $C(K).$

TL;DR

This paper investigates the property of certain continua in relation to double commutants of multiplication operators on $C(K)$, identifying classes of continua with this property and providing new examples.

Contribution

It extends the class of continua known to have the double commutant property, including arc connected but not locally connected continua, and constructs a non-arc connected example.

Findings

Locally connected metrizable continua are in $ ext{DCP}$.

Arc connected but not locally connected continua can be in $ ext{DCP}$.

A non-arc connected continuum in $ ext{DCP}$ is constructed.

Abstract

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 $f \in C(K)$ 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 $K \in \mathcal{DCP}$. It was proved in \cite{Ki} that any locally connected metrizable continuum is in $\mathcal{DCP}$. In this paper we indicate a class of arc connected but not locally connected continua that are in $\mathcal{DCP}$. We also construct an example of a continuum that is not arc connected but is in $\mathcal{DCP}$.