LAUSR

Let $$\mathcal {A}$$ and $$\mathcal {B}$$ be two factor von Neumann algebras. In this paper, we proved that a bijective mapping $$\varPhi :\mathcal {A}\rightarrow \mathcal {B}$$ satisfies $$\varPhi (a\circ b-ba^{*})=\varPhi (a)\circ \varPhi (b)-\varPhi (b)\varPhi (a)^{*}$$ (where $$\circ $$ is the special Jordan product on $$\mathcal {A}$$ and $$\mathcal {B},$$ respectively), for all elements $$a,b\in \mathcal {A}$$, if and only if $$\varPhi $$ is a $$*$$-ring isomorphism. In particular, if the von Neumann algebras $$\mathcal {A}$$ and $$\mathcal {B}$$ are type I factors, then $$\varPhi $$ is a unitary isomorphism or a conjugate unitary isomorphism.