Non-linear mixed Jordan triple $1$-$*$-product on von Neumann algebras

Abstract

It is shown that if $\Ma$ and $\Na$ are two von Neumann algebras, one of which has no central abelian projection with $\psi: \Ma \rightarrow \Na$ satisfying mixed Jordan triple $1$-$*$-product, i.e., $$\psi(\Aa \circ \Ba \bullet \Ca)=\psi(\Aa) \circ \psi(\Ba) \bullet \psi(\Ca)$$ for all $\Aa, \Ba, \Ca\in \Ma$, then there exists a bijective map $\Psi: \Ma \rightarrow \Na$ such that $\Psi(\Aa)=\psi(\Ia)\psi(\Aa)$ with $\psi(\Ia)^2=\Ia$, whenever $\psi(\Ia)$ is central, and there exsit a central projection $\mathfrak{P} \in \Ma $ such that the restriction of $\psi$ to $\Ma\mathfrak{P}$ is a linear $*$-isomorphism, and to $\Ma(\Ia -\mathfrak{P})$ is a conjugate linear $*$-isomorphism.