Arens regularity and weak topological center of module actions

TL;DR

This paper introduces the concept of a weak topological center for Banach algebra modules, explores its relation to Arens regularity and reflexivity, and applies it to the study of weak amenability of Banach algebras.

Contribution

It defines the weak topological center for module actions and investigates its connections with Arens regularity, reflexivity, and weak amenability.

Findings

Relation between weak topological center and topological center of $A^{**}$

Extension of the concept to module actions and their properties

Applications to weak amenability of Banach algebras

Abstract

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and we will find some relations between this concept and the topological center of $A^{**}$. We also extend this new definition into the module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras.