Banach module valued separating maps and automatic continuity

Document Type : Research Paper



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