On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

Abstract

Abstract The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if ℛ admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x* + xyD(x) for all x,y ∈ ℛ, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y ∈ ℛ, then D is a *-derivation. Moreover this result makes it possible to prove that if ℛ satis es 2D(xn ) = D(xn− 1)x* + xn− 1 D(x) + D(x)(x*) n− 1 + xD(xn− 1) for all x ∈ ℛ and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(ℋ). In particular, we prove that if ℋ be a real or complex Hilbert space, with dim(ℋ) > 1, admitting a linear mapping D : 𝒜(ℋ) → ℬ(ℋ) (where ℬ(ℋ) stands for the bounded linear operators) such that 2 D ( A n ) = D ( A n − 1 ) A * + A n − 1 D ( A ) + D ( A ) ( A * ) n − 1 + A D ( A n − 1 ) $$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$ for all A∈𝒜(ℋ). Then D is of the form D(A) = AB−BA* for all A∈𝒜(ℋ) and some fixed B ∈ ℬ(ℋ), which means that D is Jordan *-derivation.