Maps preserving -Lie product on factor von Neumann algebras

Abstract

Let and be two factor von Neumann algebras with dimensions greater than 1. In this paper, we consider a map from onto preserving the -Lie product, that is, for all . It is proved that a bijective map from onto preserves the -Lie product, if and only if is a linear -isomorphism or a conjugate linear -isomorphism. In particular, if the factor von Neumann algebras and are and , then is a unitary isomorphism or a conjugate unitary isomorphism.