Zero Jordan and Lie product determined of certain classes of Banach algebras

In this paper, we study the notions of zero Jordan and Lie product determined of amalgamated duplication Banach algebra of [Formula: see text]. Indeed, we give a characterization of zero Lie product determined of [Formula: see text] in the case where [Formula: see text] is a symmetric algebraic Banach [Formula: see text]-bimodule. Using this, we obtain a characterization of zero Lie product determined of the class of [Formula: see text]-Lau product Banach algebras. Moreover, we study zero Jordan product determined of [Formula: see text] where [Formula: see text] equipped with a bounded approximate identity. Finally, we give a more accurate description when [Formula: see text] is a unital Banach algebra and list some open problems.