Let $\mathcal{A}$ be a unital ∗-algebra over the complex fields $\mathbb{C}$. In this article, it is proved that a nonlinear mixed bi-skew Jordan n-derivation is an additive ∗-derivation under certain conditions. As applications, the main result is applied to some special classes of ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra and von Neumann algebra with no central summands of type $I_1$.

Let $\mathcal{A}$ be a unital ∗-algebra over the complex fields $\mathbb{C}$. For any $H_1,H_2\in \mathcal{A}$, a product ${[H_1,H_2]}_{•}=H_1{H}_2-H_2{H}_1^{*}$ is called the skew Lie product. In this article, it is shown that if a map $\xi$ : $\mathcal{A}\to \mathcal{A}$ (not necessarily linear) satisfies $\xi \left(P_n(H_1,H_2,\dots ,H_n)\right)={\sum }_{i=1}^n{P}_n(H_1,\dots ,$$H_{i-1},\xi \left(H_i\right),H_{i+1},\dots ,H_n)(n\ge 3)$ for all $H_1,H_2,\dots ,H_n\in \mathcal{A}$, then $\xi$ is additive. Moreover, if $\xi \left(i\frac{e}{2}\right)$ is self-adjoint, then $\xi$ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type $I_1$.

Let 𝒜 {\mathcal{A}} be a unital ∗ {\ast} -algebra over the complex field ℂ {\mathbb{C}} . For any μ 1 , μ 2 , μ 3 , … , μ n ∈ 𝒜 {\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}\in\mathcal{A}} , a product μ 1 ∘ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 \mu_{1}\circ\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1} is called Jordan product and μ 1 ∙ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 ∗ {\mu_{1}\bullet\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1}^{\ast}} is called skew Jordan product. Define P 3 ⁢ ( μ 1 , μ 2 , μ 3 ) = μ 1 ∘ μ 2 ∙ μ 3 P_{3}(\mu_{1},\mu_{2},\mu_{3})=\mu_{1}\circ\mu_{2}\bullet\mu_{3} (called mixed Jordan triple product) and P n ⁢ ( μ 1 , μ 2 , … , μ n ) = μ 1 ∘ μ 2 ∘ ⋯ ∙ μ n P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n})=\mu_{1}\circ\mu_{2}\circ\cdots\bullet\mu% _{n} (called mixed Jordan n -product) for all integer n ≥ 3 {n\geq 3} . In this article, it is shown that a map (called nonlinear mixed Jordan n -derivation) φ : 𝒜 → 𝒜 {\varphi:\mathcal{A}\rightarrow\mathcal{A}} satisfies φ ⁢ ( P n ⁢ ( μ 1 , μ 2 , … , μ n ) ) = ∑ i = 1 n P n ⁢ ( μ 1 , … , ν i - 1 , φ ⁢ ( ν i ) , ν i + 1 , … , ν n ) {\varphi(P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n}))=\sum_{i=1}^{n}P_{n}(\mu_{1},% \ldots,\nu_{i-1},\varphi(\nu_{i}),\nu_{i+1},\ldots,\nu_{n})} for all ν 1 , ν 2 , … , ν n ∈ 𝒜 {\nu_{1},\nu_{2},\ldots,\nu_{n}\in\mathcal{A}} if and only if φ is an additive ∗ {\ast} -derivation. As applications, our main result is applied to several special classes of unital ∗ {\ast} -algebras such as prime ∗ {\ast} -algebras, factor von Neumann algebras and von Neumann algebras with no central summands of type I 1 {I_{1}} .

This paper’s major goal is to work on commutativity of σ-prime rings with second kind involution σ, involving generalized derivation satisfy the certain differential identities. Finally, we provide some examples to demonstrate that the conditions assumed in our results are not unnecessary