Bulletin of the
Korean Mathematical Society BKMS

In this paper we show that if $D$ is a continuous linear Jordan derivation on a Banach algebra $A$ satisfying $[[D(x^{n})$, $x^{n}]$, $x^{n}]$ $\in$ rad$(A)$ for a positive integer $n$ and for all $x \in A$, then $D$ maps $A$ into rad($A$).

Keywords: extended centroid, Jordan derivation, Jacobson radical

MSC numbers: 47B47, 46H99