Jordan derivations on semiprime rings

Abstract

I. N. Herstein has proved that any Jordan derivation on a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torsion free prime ring is a derivation. In this paper we prove that Herstein’s result is true in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torsion free semiprime rings. This result makes it possible for us to prove that any linear Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair in [<bold>5</bold>].