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Jordan derivations and Jordan left derivations of Banach algebras

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Abstract

In this paper we obtain some results concerning Jordan derivations and Jordan left derivations mapping into the Jacobson radical. Our main result is the following: Let $d$ be a Jordan derivation (resp. Jordan left derivation) of a complex Banach algebra $A$. If $d^{2}(x)=0$ for all $x \in A$, then we have $d(A) \subseteq rad(A)$

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