Nonlinear Mixed Bi-Skew Jordan and Bi-Skew Lie Triple Higher Derivations on *-Algebras

Let A be a complex unital *-algebra containing a nontrivial projection and satisfying a Peirce-faithfulness condition. Let D={Dn}n∈N be a family of mappings from A into itself, not assumed to be additive, with D0=idA. Suppose that Dn([A•B,C]*)=∑p+q+r=n[Dp(A)•Dq(B),Dr(C)]* for all A,B,C∈A and n∈N, where A•B=AB*+BA* and [A,B]*=AB*−BA*. We prove that every such family is additive, preserves the involution, and satisfies the Hasse–Schmidt identity Dn(AB)=∑p+q=nDp(A)Dq(B). Consequently, every nonlinear mixed bi-skew Jordan and bi-skew Lie triple higher derivation satisfying the stated hypothesis is an additive higher *-derivation. Applications are obtained for prime *-algebras, standard operator algebras, factor von Neumann algebras, and von Neumann algebras with no central summands of type I1.