Nonlinear mixed Jordan-type derivations on *-algebras

Abstract

Let 𝒜 be a unital ∗ -algebra over the complex field ℂ . For any μ 1 , μ 2 , μ 3 , … , μ n ∈ 𝒜 , a product μ 1 ∘ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 is called Jordan product and μ 1 ∙ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 ∗ is called skew Jordan product. Define P 3 ⁢ ( μ 1 , μ 2 , μ 3 ) = μ 1 ∘ μ 2 ∙ μ 3 (called mixed Jordan triple product) and P n ⁢ ( μ 1 , μ 2 , … , μ n ) = μ 1 ∘ μ 2 ∘ ⋯ ∙ μ n (called mixed Jordan n-product) for all integer n ≥ 3 . In this article, it is shown that a map (called nonlinear mixed Jordan n-derivation) φ : 𝒜 → 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 ) for all ν 1 , ν 2 , … , ν n ∈ 𝒜 if and only if φ is an additive ∗ -derivation. As applications, our main result is applied to several special classes of unital ∗ -algebras such as prime ∗ -algebras, factor von Neumann algebras and von Neumann algebras with no central summands of type I 1 .