On the structure of Gelfand transform and BED property for Abstract Segal algebras

Abstract

Let $(\mathcal A,\|\cdot\|_{\mathcal A})$ be a commutative and semisimple
Banach algebra and $(\mathcal B,\|\cdot\|_{\mathcal B})$ be an abstract Segal algebra with respect to $\mathcal A$. In this paper, we first show that $\mathcal A$
is Tauberian if and only if $\mathcal B$ is Tauberian. Then we prove that $\widehat{\mathcal A}\subseteq C_{\rm BSE}^0(\Delta(\mathcal A))$ if and only if $\widehat{\mathcal B}\subseteq C_{\rm BSE}^0(\Delta(\mathcal B))$.
Afterwards, we conclude that whenever $\mathcal A$ is a $\rm BED$ algebra,
then $\mathcal B$ is a $\rm BED$ algebra if and only if $C_{\rm BSE}^0(\Delta(\mathcal B))\subseteq \widehat{\mathcal B}$.