Zero product determined of amalgamated duplication of Banach algebras

Abstract

In this paper, we study some sufficient and necessary conditions for the amalgamated duplication Banach algebra [Formula: see text] to be zero product determined. Precisely, we show that in the case where [Formula: see text] has a bounded approximate identity, [Formula: see text] is zero product determined if and only if [Formula: see text] is zero product determined and [Formula: see text] has the property [Formula: see text]. Moreover, we give a better characterization when [Formula: see text] is unital. As an application, we also construct some examples on semidirect product of Banach algebras and the [Formula: see text]–Lau product of Banach algebras. Finally, we raise some open problems related to the zero product determined of classical module extension Banach algebras.