Disjointness-preserving linear maps on Banach function algebras associated with a locally compact group

Abstract

We introduce a certain property of commutative Banach algebras which we call property $\mathrm{O}\mathbb{B}$. We prove that every bounded disjointness-preserving linear map from a commutative Banach algebra with the aforesaid property to any semisimple, commutative Banach algebra is a weighted composition map. Further, it is shown that a variety of important Banach algebras in harmonic analysis have the property $\mathrm{O}\mathbb{B}$.