Topological centers of $n-$th dual of module actions

TL;DR

This paper investigates the topological centers of the n-th duals of Banach modules, extending existing propositions and providing new conditions for their characterization, with applications to group algebras.

Contribution

It extends propositions on topological centers to the n-th duals of Banach modules for even n, introducing new conditions for their equalities.

Findings

Topological centers equal the modules themselves under new conditions

Extensions of Lau and Ulger's propositions to n-th duals

Applications to group algebra contexts

Abstract

In this paper, we will study the topological centers of $n-th$ dual of Banach $A-module$ and we extend some propositions from Lau and \"{U}lger into $n-th$ dual of Banach $A-modules$ where $n\geq 0$ is even number. Let $B$ be a Banach $A-bimodule$. By using some new conditions, we show that ${{Z}^\ell}_{A^{(n)}}(B^{(n)})=B^{(n)}$ and ${{Z}^\ell}_{B^{(n)}}(A^{(n)})=A^{(n)}$. We also have some conclusions in group algebras.