Nonlinear maps preserving Lie products on factor von Neumann algebras

Abstract

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra M into another factor von Neumann algebra N is of the form A → ψ ( A ) + ξ ( A ) , where ψ : M → N is an additive isomorphism or the negative of an additive anti-isomorphism and ξ : M → C I is a map with ξ ( AB - BA ) = 0 for all A , B ∈ M .