Characterizing ξ -Lie Multiplicative Isomorphisms on Von Neumann Algebras

Abstract

Let $ℳ$ and $𝒩$ be von Neumann algebras without central summands of type $I_1$. Assume that $\xi \in ℂ$ with $\xi \ne 1$. In this paper, all maps $\mathrm{\Phi }:ℳ\to 𝒩$ satisfying $\mathrm{\Phi }\left(AB-\xi BA\right)=\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(B\right)-\xi \mathrm{\Phi }\left(B\right)\mathrm{\Phi }(A)$ are characterized.