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

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-module X× Y (X is a Banach A,A-module and Y is a Banach A-module) has a BSE-module property if and only if X is a BSE Banach A,A-module and Y is a BSE Banach A-module.