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 The properties of Jordan derivations of semiprime rings and Banach algebras, II Commun. Korean Math. Soc. 2019 Vol. 34, No. 3, 811-818 https://doi.org/10.4134/CKMS.c180264Published online July 31, 2019 Byung-Do Kim Gangneung-Wonju National University Abstract : Let $A$ be a Banach algebra with $\mbox{rad}(A)$. We show that if there exists a continuous linear Jordan derivation $D$ on $A$, then $$[D(x),x]D(x)^2\in \mbox{rad}(A)$$ if and only if $D(x)[D(x),x]D(x)\in \mbox{rad}(A)$ for all $x\in A$. Keywords : Jordan derivation, derivation, semiprime ring, Banach algebra, the (Jacobson) radical MSC numbers : 16N60, 16W25, 17B40 Full-Text :