The Arens Regularity And Weak Topological Center Of Module Actions

TL;DR

This paper introduces a new concept called the weak topological center for Banach algebra modules, explores its properties, and investigates its implications for Arens regularity, reflexivity, and weak amenability of Banach algebras.

Contribution

It defines the weak topological center for module actions and establishes its relationship with Arens regularity and reflexivity, extending the understanding of Banach algebra structures.

Findings

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

Connection between weak topological center and reflexivity

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 find some relations between it 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.