Nonlinear Mixed Skew Jordan-Type Higher Derivations on ∗-Algebras

Let A be a unital ∗-algebra containing a nontrivial projection P. In this paper, we introduce the notion of a nonlinear mixed skew Jordan-type derivation (that is, pnS1,S2,S3,⋯,Sn=S1⋄S2•S3•⋯•Sn, where S⋄K=S*K+K*S and S•K=SK+KS* for all S,K∈A) and show that if a map Φ: A→A satisfies Φ(pnS1,S2,⋯,Sn)=∑k=1npn(S1,S2,⋯Sk−1,Φ(Sk),Sk+1,⋯,Sn), for all S1,S2,⋯,Sn∈A and all n≥3, then Φ is an additive ∗-derivation. Furthermore, we generalize this result to higher derivable maps on A. As applications, the above result is applied to several special classes of unital ∗-algebras such as prime ∗-algebras, factor von Neumann algebras and standard operator algebras.