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In this paper we prove the following result. Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators on X , and let A ( X ) ⊂ L ( X ) be a standard operator algebra. Suppose we have a linear mapping D : A ( X ) → L ( X ) satisfying the relation D ( A 3 ) = D ( A ) A 2 + AD ( A ) A + A 2 D ( A ), for all AA ( X ). In this case D is of the form D ( A ) = ABBA , for all AA ( X ) and some BL ( X ). We apply this result, which generalizes a classical result of Chernoff, to semisimple H *-algebras.

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Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on  X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all AA(X) and some B L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.

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Let Abe a semisimple H*-algebra and let T: AAbe an additive mapping such that T(x n +1)=T(x)x n+x n T(x) holds for all xAand some integer n≥1. In this case Tis a left and a right centralizer.

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Let Rbe a 2-torsion free semiprime *-ring and let T:R?Rbe an additive mapping such that T(xx*)=T(x)x* is fulfilled for all x ?R. In this case T(xy)=T(x)yholds for all pairs x,y?R.

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We characterize certain maps by their action on a fixed polynomial in noncommuting variables on algebras satisfying certain d -freeness condition. Consequently, a characterization of the centroid of a prime ring is obtained.

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