On Mixed Nonlinear Skew Lie (Jordan) Products on Von Neumann Algebras

Abstract

Let U and V be von Neumann algebras. For D,E∈U, define by D,E∗=DE−ED∗D⋄E=DE+ED∗ the skew Lie (Jordan) product of D and E. In this manuscript, we study a bijective mapping Λ:U⟶V satisfying ΛD,E∗=ΛD⋄ΛE for all D,E∈U. Especially, we elaborate on the characteristics of the map Λ. The study revealed several constructive facts, leading to the conclusion that the map Λ is additive.