Banach module valued separating maps and automatic continuity

Abstract

For two algebras $A$ and $B$, a linear map $T:A longrightarrow <br />B$ is called separating, if $xcdot y=0$ implies $Txcdot Ty=0$ for <br />all $x,yin A$. The general form and the automatic continuity of <br />separating maps between various Banach algebras have been studied <br />extensively. In this paper, we first extend the notion of separating <br />map for module case and then we give a description of a linear <br />separating map $T:B longrightarrow X$, where $B$ is a unital <br />commutative semisimple regular Banach algebra satisfying the <br />Ditkin's condition and $X$ is a left Banach module over a unital <br />commutative Banach algebra. We also show that if $X$ is hyper <br />semisimple and $T$ is bijective, then $T$ is automatically <br />continuous and $T^{-1}$ is separating as well.