Communications of the
Korean Mathematical Society CKMS

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 $d(A) \subseteq rad(A)$

Keywords: Banach algebra, Jordan derivation, Jordan left derivation, Jacobson radical

MSC numbers: Primary 46H99; Secondary 47B47