Linear maps preserving zero products on nest subalgebras of von Neumann algebras

In this paper, it is proved that every surjective linear map preserving identity and zero products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is an isomorphism; and that every surjective weakly continuous linear map preserving identity and zero Jordan products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is either an isomorphism or an anti-isomorphism.