Arens Regularity Of Bilinear Forms And Unital Banach Module Spaces

TL;DR

This paper investigates the relationships between Arens regularity of bilinear forms and the module structures of Banach algebras and their duals, establishing conditions for factorization and unital properties.

Contribution

It introduces new criteria linking Arens regularity, module factorization, and unital structures in Banach algebras and their duals.

Findings

Bilateral regularity of mappings relates to the regularity of the underlying Banach algebras.

Factorization of Banach modules is characterized by the unitality of their second duals.

Conditions are established under which dual modules factor on both sides with respect to the algebra.

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 $A^{**}-module$.