Some Remarks on Derivations in Semiprime Rings and Standard Operator Algebras

In this paper identities related to derivations on semiprime rings and standard operator algebras are investigated. We prove the following result which generalizes a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊆ L(X) be a standard operator algebra. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(A3) = D(A2)A+A2D(A) +D(A)A2 +AD(A2) for all A ∈ A(X). In this case D is of the form D(A) = AB − BA for all A ∈ A(X) and some fixed B ∈ L(X), which means that D is a linear derivation. This research has been motivated by the work of Brešar ([3]) and Chernoff ([4]) and it is a continuation of our recent work ([11–13]). Throughout, R will represent an associative ring with center Z(R). As usual we write [x, y] for xy − yx. Given an integer n ≥ 2, a ring R is said to be n−torsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies a = 0 or b = 0, and semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B → A is called a linear derivation in case D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. In case we have a ring R an additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R such that D(x) = [x, a] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in 2010 Mathematics Subject Classification. 16W10, 46K15, 39B05.