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 $B^{**}$ is an unital $A^{**}-module$, and we define locally topological center for elements of $A^{**}$ and will show that when locally topological center of mixed unit of $A^{**}$ is $B^{**}$, then $B^*$ factors on both sides with respect to $A$ if and only if $B^{**}$ has a unit as $A^{**}-module$.