Lie $^*$-triple homomorphisms into von Neumann algebras

Abstract:

Let $M$ and $N$ be associative $\ast$-algebras. A Lie $\ast$-triple homomorphism of $M$ into $N$ is a $\ast$-linear map $\phi :M \to N$ such that \[ \phi [[A,B],C] = [[\phi (A),\phi (B)],\phi (C)].\] (Here $M$ and $N$ are considered as Lie $\ast$-algebras with $[X,Y] = XY - YX.)$ In this note we prove that if $N$ is a von Neumann algebra with no central abelian projections and if $\phi$ is onto, there exists a central projection $D$ in $N$ such that $D\phi$ is a Lie $\ast$-homomorphism of $[M,M]$, and $(I - D)\phi$ is a Lie $\ast$-antihomomorphism of $[M,M]$.