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

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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 admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y, then D is a *-derivation. Moreover this result makes it possible to prove that if satis es 2D(xn) = D(xn−1)x* + xn−1D(x) + D(x)(x*)n−1 + xD(xn−1) for all x 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(). In particular, we prove that if be a real or complex Hilbert space, with dim() > 1, admitting a linear mapping D : 𝒜() → ℬ() (where () stands for the bounded linear operators) such that

2D(An)=D(An1)A*+An1D(A)+D(A)(A*)n1+AD(An1)

for all A𝒜(). Then D is of the form D(A) = AB−BA* for all A𝒜() and some fixed B(), which means that D is Jordan *-derivation.

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