Factorization property and Arens regularity

TL;DR

This paper investigates the Arens regularity of module actions in Banach algebras, extending existing propositions and establishing conditions for factorization and topological centers in dual spaces.

Contribution

It generalizes previous results on Arens regularity, introduces new relationships between topological centers, and extends definitions of multipliers for module actions.

Findings

Established relationships between topological centers of duals.

Provided necessary and sufficient conditions for factorization.

Extended definitions of multipliers for module actions.

Abstract

In this paper, we study the Arens regularity properties of module actions and we extend some proposition from Baker, Dales, Lau and others into general situations. For Banach $A-bimodule$ $B$, let $Z_1(A^{**})$, ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of second dual of Banach algebra $A$, left module action $\pi_\ell:~A\times B\rightarrow B$ and right module action $\pi_r:~B\times A\rightarrow B$, respectively. We establish some relationships between them and factorization properties of $A^*$ and $B^*$. We search some necessary and sufficient conditions for factorization of $A^*$, $B$ and $B^*$ with some results in group algebras. We extend the definitions of the left and right multiplier for module actions.