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 $\eta$ be a non-zero complex number. A nonlinear bijective map $\phi:\mathcal A\rightarrow\mathcal B$ has been demonstrated to satisfy $$\phi([A,B]{*}^{\eta}\diamond{\eta} C)=[\phi(A),\phi(B)]{*}^{\eta}\diamond{\eta}\phi(C)$$ for all $A,B,C\in\mathcal A.$ If $\eta=1,$ then $\phi$ is a linear $$-isomorphism, a conjugate linear $$-isomorphism, the negative of a linear $$-isomorphism, or the negative of a conjugate linear $$-isomorphism. If $\eta\neq 1$ and satisfies $\phi(I)=1,$ then $\phi$ is either a linear $$-isomorphism or a conjugate linear $$-isomorphism.