A note on strong skew Jordan product preserving maps on von Neumann algebras

Abstract

Let \(\mathcal {A}\) be a von Neumann algebra acting on the complex Hilbert space \(\mathcal {H}\) and \(\Phi {:}\,\mathcal {A} \longrightarrow \mathcal {A}\) be a surjective map that satisfies the condition $$\begin{aligned} \Phi (T)\Phi (P)+\Phi (P)\Phi (T)^*=TP+PT^* \end{aligned}$$ for all T and all projections P in \(\mathcal {A}\). We characterize the concrete form of \(\Phi \) on selfadjoint elements of \(\mathcal {A}\). Also when \(\mathcal {A}\) is a factor von Neumann algebra, it is shown that \(\Phi \) is either of the form \(\Phi (T)=T+i\tau (T) I\) or of the form \(\Phi (T)=-T+i\tau (T)I\), where \(\tau {:}\,\mathcal {A}\longrightarrow \mathbb {R}\) is a real map.