Nonlinear maps preserving the mixed Jordan triple $η$-$*$-product between factors

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and $η$ be a non-zero complex number. A nonlinear bijective map $ϕ:\mathcal A\rightarrow\mathcal B$ has been demonstrated to satisfy $$ϕ([A,B]_{*}^η\diamond_η C)=[ϕ(A),ϕ(B)]_{*}^η\diamond_ηϕ(C)$$ for all $A,B,C\in\mathcal A.$ If $η=1,$ then $ϕ$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $η\neq 1$ and satisfies $ϕ(I)=1,$ then $ϕ$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.