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

Let (A, \| · \|A) be a commutative and semisimple Banach algebra and (B, \| · \|B) be an abstract Segal algebra with respect to A . In this paper, we first show that A is Tauberian if and only if B is Tauberian. Then we prove that A ⊆ C_0BSE(Δ(A)) and only if B ⊆ C_0BSE(Δ(B)). After wards, we conclude that whenever A is a B BED algebra, then B is a BED algebra if and only if C_0BSE(Δ(B)) ⊆ B.$.