Characterization and automatic continuity of separating maps between Banach modules

Abstract

A linear map $T:\mathcal{A}\to \mathcal{B}$ between algebras (or spaces of functions) $\mathcal{A}$ and $\mathcal{B}$ is called separating if $x \cdot y=0$ implies $Tx\cdot Ty=0$ for all $x,y\in \mathcal{A}$. It is well known that a separating map between certain commutative semisimple Banach algebras is very close to being a weighted composition operator on the maximal ideal spaces. In this paper, after introducing the notion of the cozero set for the elements of a Banach module, we first extend the notion of the separating maps to Banach module case. Our approach depends on the notion of point multipliers on a Banach module $\mathcal{X}$ and the relation between hyper maximal submodules of $\mathcal{X}$ and point multipliers on it. Then we generalize some well known results about separating maps between certain subspaces of continuous functions to Banach module case. In particular, we show that, imposing some additional assumptions on Banach modules, such map can be represented as a variation of a weighted composition operator. We also obtain a result concerning the automatic continuity of a bijective separating map whose inverse is also separating.