Approximate Identity and Arens Regularity of Some Banach Algebras

TL;DR

This paper investigates the conditions under which Banach algebras with approximate identities are Arens regular and introduces new properties that characterize this regularity, with applications to classical group algebras.

Contribution

It introduces the $LW^*W$ and $RW^*W$ properties, providing new criteria for Arens regularity in Banach algebras with approximate identities.

Findings

Arens regularity linked to $LW^*W$ and $RW^*W$ properties.

Equivalence of unitality of $A^{**}$ and existence of $W^*BAI$.

Applications to group algebras like $l^1(G)$ and $L^1(G)$.

Abstract

Let $A$ be a Banach algebra with the second dual $A^{**}$. If $A$ has a bounded approximate identity $(=BAI)$, then $A^{**}$ is unital if and only if $A^{**}$ has a $weak^* bounded approximate $$identity(=W^*BAI)$. If $A$ is Arens regular and $A$ \noindent has a BAI, then $A^*$ factors on both sides. In this paper we introduce new concepts $LW^*W$ and $RW^*W$- property and we show that under certain conditions if $A$ has $LW^*W$ and $RW^*W$- property, then $A$ is Arens regular and also if $A$ is Arens regular, then $A$ has $LW^*W$ and $RW^*W$- property. We also offer some applications of these new concepts for the special algebras $l^1(G), L^1(G), M(G)$, and $A(G)$.