The topological centers and factorization properties of module actions and $\ast-involution$ algebras

TL;DR

This paper investigates the topological centers and factorization properties of module actions and $ ext{*}$-involution algebras in Banach modules, extending existing theories and introducing new properties with applications to group algebras.

Contribution

It extends propositions on module actions, introduces new properties like $Lw^*w$ and $Rw^*w$, and explores their implications for Arens regularity and $ ext{*}$-involution algebras.

Findings

Established relationships between topological centers of module actions.

Introduced $Lw^*w$ and $Rw^*w$ properties for Banach bimodules.

Applied results to group algebras.

Abstract

For Banach left and right module actions, we extend some propositions from Lau and $\ddot{U}lger$ into general situations and we establish the relationships between topological centers of module actions. We also introduce the new concepts as $Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we obtain some conclusions in the topological center of module actions and Arens regularity of Banach algebras. we also study some factorization properties of left module actions and we find some relations of them and topological centers of module actions. For Banach algebra $A$, we extend the definition of $\ast-involution$ algebra into Banach $A-bimodule$ $B$ with some results in the factorizations of $B^*$. We have some applications in group algebras.