Nonlinear Mixed Skew Lie-Type Derivations on \({}\)-Algebras

Consider A as a unital \({}\)-algebra. Given elements A,B∈A, the operations A•B=AB+BA* and [A,B]*=AB−BA* represent the skew Jordan product and the skew Lie product, correspondingly. Within this study, we demonstrate that a nonlinear mapping Φ:A→A adhering to Φ[A1•A2•…•An−1,An]*=∑j=1n[A1•A2•…•Aj−1•Φ(Aj)•Aj+1•…•An−1,An]* for every A1,A2,…,An∈A with n≥3, then Φ is an additive \({}\)-derivation. As an application of our main results, we establish characterizations of prime \({}\)-algebras, standard operator algebras, von Neumann algebras without central summands of type I1, and factor von Neumann algebras.