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 $\A\rtimes\U$-module $X\times Y$ ($X$ is a Banach $\A,\U$-module and $Y$ is a Banach $\U$-module) has a BSE-module property if and only if $X$ is a BSE Banach $\A,\U$-module and $Y$ is a BSE Banach $\U$-module.