Let $ \mathfrak{A} $ be a unital $ * $-algebra with identity $ \mathscr{I} $. For $ \mathscr{A}, \mathscr{B}\in\mathfrak{A} $, define the bi-skew Jordan product
$ \mathscr{A}\bullet \mathscr{B} = \mathscr{A}\mathscr{B}^{*}+\mathscr{B}\mathscr{A}^{*} $
and the bi-skew Lie product
$ [\mathscr{A},\mathscr{B}]_{\diamond} = \mathscr{A}\mathscr{B}^{*}-\mathscr{B}\mathscr{A}^{*}. $
Suppose that a nonlinear mapping $ \Phi:\mathfrak{A}\to\mathfrak{A} $ satisfies
$ \Phi\!\left(\big[\mathscr{A}_1\bullet\mathscr{A}_2\bullet\cdots\bullet\mathscr{A}_{n-1},\mathscr{A}_n\big]_{\diamond}\right) = \sum\limits_{j = 1}^{n}\big[\mathscr{A}_1\bullet \cdots \bullet\mathscr{A}_{j-1}\bullet \Phi(\mathscr{A}_j)\bullet\mathscr{A}_{j+1}\bullet \cdots\bullet\mathscr{A}_{n-1},\mathscr{A}_n\big]_{\diamond} $
for all suitable elements $ \mathscr{A}_1, \ldots, \mathscr{A}_n\in\mathfrak{A} $, where $ \mathscr{A}_1, \mathscr{A}_2\in\{\mathscr{I}, i\mathscr{I}\} $ and $ \mathscr{A}_j = \mathscr{I} $ for every $ j = 3, 4, \ldots, n-2 $. We prove that $ \Phi $ is an additive $ * $-derivation on $ \mathfrak{A} $. As applications, several consequences are obtained for prime $ * $-algebras, factor von Neumann algebras, von Neumann algebras without central summands of type $ \mathscr{I}_1 $, and standard operator algebras. Moreover, a conjecture is proposed to motivate further research in this direction. The obtained results extend and generalize a recent result of Abbasi et al. [Non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $ * $-algebras, Ricerche Mat., 2026.] concerning non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $ * $-algebras.
Citation: Abu Zaid Ansari, Mohammad Shane Alam, Nof T. Alharbi, Ishraga A. Mohamed. A note on nonlinear mixed bi-skew Jordan and bi-skew Lie $ n $-derivations on $ * $-algebras[J]. AIMS Mathematics, 2026, 11(6): 16936-16951. doi: 10.3934/math.2026693
Let $ \mathfrak{A} $ be a unital $ * $-algebra with identity $ \mathscr{I} $. For $ \mathscr{A}, \mathscr{B}\in\mathfrak{A} $, define the bi-skew Jordan product
$ \mathscr{A}\bullet \mathscr{B} = \mathscr{A}\mathscr{B}^{*}+\mathscr{B}\mathscr{A}^{*} $
and the bi-skew Lie product
$ [\mathscr{A},\mathscr{B}]_{\diamond} = \mathscr{A}\mathscr{B}^{*}-\mathscr{B}\mathscr{A}^{*}. $
Suppose that a nonlinear mapping $ \Phi:\mathfrak{A}\to\mathfrak{A} $ satisfies
$ \Phi\!\left(\big[\mathscr{A}_1\bullet\mathscr{A}_2\bullet\cdots\bullet\mathscr{A}_{n-1},\mathscr{A}_n\big]_{\diamond}\right) = \sum\limits_{j = 1}^{n}\big[\mathscr{A}_1\bullet \cdots \bullet\mathscr{A}_{j-1}\bullet \Phi(\mathscr{A}_j)\bullet\mathscr{A}_{j+1}\bullet \cdots\bullet\mathscr{A}_{n-1},\mathscr{A}_n\big]_{\diamond} $
for all suitable elements $ \mathscr{A}_1, \ldots, \mathscr{A}_n\in\mathfrak{A} $, where $ \mathscr{A}_1, \mathscr{A}_2\in\{\mathscr{I}, i\mathscr{I}\} $ and $ \mathscr{A}_j = \mathscr{I} $ for every $ j = 3, 4, \ldots, n-2 $. We prove that $ \Phi $ is an additive $ * $-derivation on $ \mathfrak{A} $. As applications, several consequences are obtained for prime $ * $-algebras, factor von Neumann algebras, von Neumann algebras without central summands of type $ \mathscr{I}_1 $, and standard operator algebras. Moreover, a conjecture is proposed to motivate further research in this direction. The obtained results extend and generalize a recent result of Abbasi et al. [Non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $ * $-algebras, Ricerche Mat., 2026.] concerning non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $ * $-algebras.
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Abu Zaid Ansari, Mohammad Shane Alam, Nof T. Alharbi, Ishraga A. Mohamed. A note on nonlinear mixed bi-skew Jordan and bi-skew Lie $ n $-derivations on $ * $-algebras[J]. AIMS Mathematics, 2026, 11(6): 16936-16951. doi: 10.3934/math.2026693
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