Abstract : The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let $A$ be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A\to A$ such that $D(x)^3[D(x),x]\in \mbox{rad}(A)$ for all $x\in A.$ In this case, we show that $D(A)\subseteq \mbox{rad}(A).$
Keywords : prime and semiprime ring, (Jacobson) radical, Jordan derivation
