The Bochner-Schoenberg-Eberlein module property for amalgamated duplication of Banach algebras

Abstract

The Bochner-Schoenberg-Eberlein module property on commutative Banach algebras is a property related to extensions of multipliers on Banach algebras to module morphisms from Banach algebras into Banach modules. In this paper, we answer the problem (1) raised in [J. Algebra Appl., 21(8) (2022), 2250155, DOI: 10.1142/S0219498822501559]. We show that the Banach $\mathcal{A}\rtimes\mathfrak{A}$-module $X\times Y$ ($X$ is a Banach $\mathcal{A},\mathfrak{A}$-module and $Y$ is a Banach $\mathfrak{A}$-module) has a BSE-module property if and only if $X$ is a BSE Banach $\mathcal{A},\mathfrak{A}$-module and $Y$ is a BSE Banach $\mathfrak{A}$-module.

Keywords

• amalgamated Banach algebra
• BSE-Banach algebra
• BSE module property
• character
• multiplier

References

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