Linear Generalized N-derivations on C-algebras*

Let n ≥ 2 be a fixed integer and A be a C^*-algebra. A permuting n-linear map G : A ⁿ → A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D : A ⁿ → A such that $ G (x₁, x₂, …, xᵢ xᵢ^{}, …, xₙ)= G (x₁, x₂, …, xᵢ, …, xₙ) xᵢ^{}+xᵢ D (x₁, x₂,…, xᵢ^{}, …, xₙ)$ holds for all $xᵢ, xᵢ^{} ∈ A $. In this paper, we investigate the structure of $C^*$-algebras involving generalized linear $n$-derivations. Moreover, we describe the forms of traces of linear $n$-derivations satisfying certain functional identity.