%0 Journal Article
%T Banach module valued separating maps and automatic continuity
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Mousavi, L.
%A Sady, F.
%D 2011
%\ 12/15/2011
%V 37
%N No. 4
%P 127-139
%! Banach module valued separating maps and automatic continuity
%K Banach algebras
%K Banach modules
%K separating maps
%K cozero set
%K point multiplier
%K Automatic continuity
%R
%X 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.
%U http://bims.iranjournals.ir/article_375_1bbfb6fc763d4e986298a5e63f05fd4c.pdf