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

Abstract

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

Keywords

• prime ring
• semiprime ring
• normal ring
• involution
• left ∗-centralizer
• Jordan left ∗-centralizer
• generalized derivation

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