Nonlinear \(*\)-Lie \(n\)-tuple derivations on prime \(*\)-algebras (Q2184232)

Let \(A \diamond B=A \cdot B-B \cdot A^{*}\) denote the \(*\)-Lie product. A~map \(\Phi : A \rightarrow A\) is said to preserve \(*\)-Lie \(n\)-tuple derivations if \[ \begin{multlined} \Phi (A_1 )\diamond A_2 \diamond \cdots \diamond A_n = \Phi (A_1) \diamond A_2 \diamond \cdots \diamond A_n + A_1 \diamond \Phi (A_2) \diamond \cdots \diamond A_n \\ + A_1 \diamond A_2 \diamond \cdots A_{n-1} \diamond \Phi (A_n) \end{multlined} \] for all \(A_1, A_2,\dots, A_n\) in \(A\). In this paper, the authors prove that such maps are automatically additive in the case where \(A\) is prime \(*\)-algebra or a von Neumann algebra without central abelian projections. This extends the same result already proven for factor von Neumann algebras in the cases \(n=2\) [\textit{W.-Y. Yu} and \textit{J.-H. Zhang}, Linear Algebra Appl. 437, No. 8, 1979--1991 (2012; Zbl 1263.46058)] and \(n=3\) [\textit{C. J. Li} et al., Acta Math. Sin., Engl. Ser. 32, No. 7, 821--830 (2016; Zbl 1362.47025)].