Abstract
In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let <TEX>$\delta$</TEX> be a spectrally bounded left Jordan derivation on a Banach algebra A. Then <TEX>$\delta$</TEX> maps A into its Jacobson radical. (ii) Let <TEX>$\delta$</TEX> be a left Jordan derivation on a unital Banach algebra A with the condition sup{r<TEX>$(c^{-1}\delta(c))$</TEX> : c <TEX>$\in$</TEX> A invertible} < <TEX>$\infty$</TEX>. Then <TEX>$\delta$</TEX> maps A into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in [2, p. 260]: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.
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