Communications of the
Korean Mathematical Society CKMS

The purpose of this paper is to obtain Jordan left derivations that map into the Jacobson radical: (i) Let $d$ be a spectrally bounded Jordan left derivation on a Banach algebra $A$. If $[d(x),x] \in rad(A)$ for all $x \in A$, then $d(A) \subseteq rad(A)$. (ii) Let $d$ be a Jordan left derivation on a unital Banach algebra $A$ with the condition sup$\{r(z^{-1}d(z))|z \in A \text{ invertible} \}< \infty$. If $[d(x),x] \in rad(A)$ for all $x \in A$, then $d(A) \subseteq rad(A)$.

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

MSC numbers: Primary 46H99; Secondary 47B47