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.