BSE-Properties of Second Dual of Banach Algebras

TL;DR

This paper investigates the relationship between the BSE-property of a commutative semisimple Arens regular unital Banach algebra and its second duals, establishing conditions under which these properties are equivalent or related.

Contribution

It provides new insights into how the BSE-property and BSE-norm property of a Banach algebra relate to those of its second dual, including conditions for equivalence.

Findings

If $A$ is a BSE-algebra, then $A^{**}$ is also a BSE-algebra.

The opposite correlation holds under certain conditions.

If $A^{**}$ is semisimple, then both $A$ and $A^{**}$ are BSE-norm algebras.

Abstract

Let $A$ be a commutative semisimple Arens regular unital Banach algebra. The correlation between the BSE-property of the Banach algebra $A$ and its second duals are assessed. It is found and approved that if $A$ is a BSE-algebra, then so is $A^{**}$. The opposite correlation will hold in certain conditions. The correlation of the BSE-norm property of the Banach algebra and its second dual are assessed and examined. It is revealed that, if $A$ is a commutative Arens regular unital Banach algebra where $A^{**}$ is semisimple, then, $A$ and $A^{**}$ are BSE-norm algebra.