LEFT JORDAN DERIVATIONS ON BANACH ALGEBRAS AND RELATED MAPPINGS
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In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let δ be a spectrally bounded left Jordan derivation on a Banach algebra А. Then δ maps Аinto its Jacobson radical. (ii) Let δ be a left Jordan derivation on a unital Banach algebra А with the condition sup$\{r(c-1δ(c)): c ∈ A invertible} < ∞. Then δ maps А into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in \cite[p.\,260]{Ash08}: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.
Abstract
In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let δ be a spectrally bounded left Jordan derivation on a Banach algebra А. Then δ maps Аinto its Jacobson radical. (ii) Let δ be a left Jordan derivation on a unital Banach algebra А with the condition sup$\{r(c-1δ(c)): c ∈ A invertible} < ∞. Then δ maps А into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in \cite[p.\,260]{Ash08}: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.
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