Let $ \mathfrak{B} $ denote a unital prime $ \ast $-algebra over the complex field $ \mathbb{C} $, and let $ \xi \in \mathbb{C} \setminus \{0, \pm 1\} $. This paper characterized a family $ \Psi = \{\varphi_m\}_{m \in \mathbb{N}} $ of maps (not necessarily additive) from $ \mathfrak{B} $ into itself that satisfy the functional identity
$ \varphi_m(A\diamond_\xi B\diamond_\xi C) = \sum\limits_{i+j+k = m}\varphi_i(A)\diamond_\xi \varphi_j(B)\diamond_\xi \varphi_k(C) $
for all $ A, B, C \in \mathfrak{B} $, where $ A \diamond_\xi B = AB\; +\; \xi BA^\ast $. It was proved that $ \Psi $ satisfies the above identity if and only if $ \Psi $ is an additive higher $ \ast $-derivation and $ \varphi_m(\xi A) = \xi \varphi_m(A) $ holds for every $ A \in \mathfrak{B} $. As an application, the result not only generalizes the structure of nonlinear $ \xi $-skew-Jordan triple derivations on prime $ \ast $-algebras, but also yields a description of nonlinear $ \xi $-skew-Jordan triple higher derivations on several important operator algebras, including standard operator algebras and factor von Neumann algebras.
Citation: Xinfeng Liang, Ying Ning. Nonlinear $ \xi $-skew-Jordan triple higher derivations on prime $ \ast $-algebras[J]. Electronic Research Archive, 2026, 34(4): 2462-2480. doi: 10.3934/era.2026113
Let $ \mathfrak{B} $ denote a unital prime $ \ast $-algebra over the complex field $ \mathbb{C} $, and let $ \xi \in \mathbb{C} \setminus \{0, \pm 1\} $. This paper characterized a family $ \Psi = \{\varphi_m\}_{m \in \mathbb{N}} $ of maps (not necessarily additive) from $ \mathfrak{B} $ into itself that satisfy the functional identity
$ \varphi_m(A\diamond_\xi B\diamond_\xi C) = \sum\limits_{i+j+k = m}\varphi_i(A)\diamond_\xi \varphi_j(B)\diamond_\xi \varphi_k(C) $
for all $ A, B, C \in \mathfrak{B} $, where $ A \diamond_\xi B = AB\; +\; \xi BA^\ast $. It was proved that $ \Psi $ satisfies the above identity if and only if $ \Psi $ is an additive higher $ \ast $-derivation and $ \varphi_m(\xi A) = \xi \varphi_m(A) $ holds for every $ A \in \mathfrak{B} $. As an application, the result not only generalizes the structure of nonlinear $ \xi $-skew-Jordan triple derivations on prime $ \ast $-algebras, but also yields a description of nonlinear $ \xi $-skew-Jordan triple higher derivations on several important operator algebras, including standard operator algebras and factor von Neumann algebras.
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Xinfeng Liang, Ying Ning. Nonlinear $ \xi $-skew-Jordan triple higher derivations on prime $ \ast $-algebras[J]. Electronic Research Archive, 2026, 34(4): 2462-2480. doi: 10.3934/era.2026113
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