THE JORDAN DERIVATIONS OF SEMIPRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS

Abstract

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation D : R ! R such that [[D(x); x]; x]D(x) = 0 or D(x)[[D(x); x]; x] = 0 for all x 2 R: In this case we have [D(x); x]3 = 0 for all x 2 R: Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A ! A such that [[D(x); x]; x]D(x) 2 rad(A) or D(x)[[D(x); x]; x] 2 rad(A) for all x 2 A: In this case, we show that D(A) (cid:18) rad(A):

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