Let U and V be locally convex algebras and f : U → V be a continuous linear map satisfying x, y ∈ U , xy = 0 ⇒ f ([x, y]) = [f (x), f (y)]. In this paper, we show on a class of locally convex algebras U that f is a Lie homomor- phism. Also, we provide a characterization of Lie derivations through zero products on this class of locally convex algebras. Finally, we use our main results for von Neumann algebras, standard operator algebras and nest algebras.
