The higher duals of certain class of Banach algebras

TL;DR

This paper investigates the properties of a specific class of Banach algebras formed by a fixed functional, focusing on their duals and properties like Arens regularity, weak amenability, and semi-simplicity.

Contribution

It introduces a new class of Banach algebras $_\varphi A$ based on a fixed functional and analyzes their higher duals and key algebraic properties.

Findings

$_\varphi A$ can be Arens regular under certain conditions

The algebra's $n$-weak amenability is characterized

Semi-simplicity depends on the properties of $\varphi$

Abstract

Given a Banach space $A$ and fix a non-zero $\varphi\in A^*$ with $\|\varphi\|\leq 1$. Then the product $a\cdot b=\langle\varphi, a\rangle\ b$ turning $A$ into a Banach algebra which will be denoted by $_\varphi A.$ Some of the main properties of $_\varphi A$ such as Arens regularity, $n$-weak amenability and semi-simplicity are investigated.