Zero product determined some nest algebras

Let AlgN be a nest algebra associated with the nest N on a (real or complex) Hilbert space H. We say that AlgN is zero product determined if for every linear space N and every bilinear map ϕ:AlgN×AlgN→V the following holds: if ϕ(A,B) = 0 whenever AB = 0, then there exists a linear map T such that ϕ(A,B)=T(AB) for all A,B∈AlgN. If we replace in this definition the ordinary product by the Jordan (resp., Lie) product, then we say that AlgN is zero Jordan (resp., Lie) product determined. We show that any finite nest algebra over a complex Hilbert space is zero product determined, and it is also zero Jordan product determined. Moreover, we show that any finite-dimensional nest algebra on a (real or complex) Hilbert space is zero Lie (resp., associative, Jordan) product determined. In addition, we characterize separately strongly operator topology continuous bilinear map ϕ from AlgN ×  AlgN into a topological linear space V with the property that ϕ(A,B)=0 whenever AB = 0 or ϕ(A,B)=0 whenever AB + BA = 0.