What question did this study set out to answer?

The aim is to explore the relationship between the Tauberian properties and BED properties of commutative semisimple Banach algebras and abstract Segal algebras.

May 19, 2026Open Access

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

Key Points

The aim is to explore the relationship between the Tauberian properties and BED properties of commutative semisimple Banach algebras and abstract Segal algebras.
Examined the Tauberian property correlation between the Banach algebra and abstract Segal algebra.
Proved inclusion conditions for the spaces C_0^{BSE} related to both algebras.
Analyzed the relationship of BED algebras between the two structures.
Established that the Banach algebra is Tauberian if and only if the abstract Segal algebra is Tauberian.
Showed that the inclusion C_0^{BSE}(()) holds under similar conditions for both algebras.
Concluded that if the Banach algebra is a BED algebra, then the abstract Segal algebra is also a BED algebra under certain inclusion conditions.

Abstract

Let (A, \| \|₀) be a commutative and semisimple Banach algebra and (B, \| \|₁) 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 C₀^BSE ( (A) ) and only if B C₀^BSE ( (B) ). After wards, we conclude that whenever A is a B BED algebra, then B is a BED algebra if and only if C₀^BSE ( (B) ) B. .

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

Key Points

Abstract

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