Let $$\mathcal{A}$$ be a prime $$\ast$$ -algebra. In this paper, assuming that $$\Phi:\mathcal{A}\to\mathcal{A}$$ satisfies $$\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B \diamond C+A\diamond\Phi(B) \diamond C+A \diamond B \diamond \Phi(C)$$ where $$A\diamond B = A^{*}B + B^{*}A$$ for all $$A,B\in\mathcal{A}$$ , we prove that $$\Phi$$ is additive an $$\ast$$ -derivation.

中文翻译：

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非线性三重积 $$A^{*}B + B^{*}A$$ 用于 $$\ast$$-代数的推导

令 $$\mathcal{A}$$ 是素数 $$\ast$$ -代数。本文假设$$\Phi:\mathcal{A}\to\mathcal{A}$$满足$$\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B \diamond C+A\diamond\Phi(B) \diamond C+A \diamond B \diamond \Phi(C)$$ 其中 $$A\diamond B = A^{*}B + B^{*}A$$对于所有 $$A,B\in\mathcal{A}$$ ，我们证明 $$\Phi$$ 是 $$\ast$$ 的可加性推导。