Mappings preserving sum of products \(a\circ b+ba^{*}\) on factor von Neumann algebras

Abstract

Let \(\mathcal{A}\) and \(\mathcal{B}\) be two factor von Neumann algebras. In this paper, we proved that a bijective mapping \(\Phi :\mathcal{A}\rightarrow \mathcal{B}\) satisfies \(\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}\) (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 \(\Phi \) is a \(\ast \)-ring isomorphism. In particular, if the von Neumann algebras \(\mathcal{A}\) and \(\mathcal{B}\) are type I factors, then \(\Phi \) is a unitary isomorphism or a conjugate unitary isomorphism.