Nonlinear maps preserving Lie products on factor von Neumann algebras (Q927744)

Let \(\mathcal A\) and \(\mathcal B\) be factor von Neumann algebras and \(\Phi : \mathcal A\to \mathcal B\) a bijective map satisfying \(\Phi(AB - BA) = \Phi(A)\Phi(B) -\Phi(B)\Phi(A)\) for all \(A, B \in \mathcal A\). It is shown that \(\Phi\) has the form \(\Phi(A) = \Psi(A) +\xi (A)\) for every \(A\in\mathcal A\), where \(\Psi\) is an additive isomorphism or the negative of an additive anti-isomorphism and \(\xi:\mathcal A\to \mathbb CI\) is a map satisfying \(\xi(AB - BA) = 0\) for all \(A, B\in \mathcal A\). Here, \(\mathbb C\) denotes the field of complex numbers.