Arens Regularity Of Bilinear Forms And Unital Banach Module Spaces

Abstract

Assume that $A$, $B$ are Banach algebras and $m:A\times B\to B$, $m^{\prime }:A\times A\to B$ are bounded bilinear mappings. We will study the relation between Arens regularities of $m$, $m^{\prime }$ and the Banach algebras $A$, $B$. For Banach $A-bimodule$ $B$, we show that $B$ factors with respect to $A$ if and only if $B^{\ast \ast }$ is an unital $A^{\ast \ast }-module$, and we define locally topological center for elements of $A^{\ast \ast }$ and will show that when locally topological center of mixed unit of $A^{\ast \ast }$ is $B^{\ast \ast }$, then $B^{\ast }$ factors on both sides with respect to $A$ if and only if $B^{\ast \ast }$ has a unit as $A^{\ast \ast }-module$.