The Arens Regularity and Weak Topological Center of Module Actions

TL;DR

This paper introduces a new concept of weak topological centers in Banach algebra modules, explores their properties, and examines implications for reflexivity, Arens regularity, and weak amenability of Banach algebras.

Contribution

It extends the definition of weak topological centers to module actions and investigates their relationships with algebraic properties like reflexivity and Arens regularity.

Findings

Defined the weak topological center of module actions.

Established relations between weak topological centers and algebra properties.

Applied the concept to analyze 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 find some relations between it and the topological center of $A^{**}$. We also extend this new definition into 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.