Left Jordan Derivations on Banach Algebras and Related Mappings

Abstract

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 A. Then δ maps A into its Jacobson radical. (ii) Let δ be a left Jordan derivation on a unital Banach algebra A with the condition sup{r(c −1 δ(c)) : c ∈ A invertible} < ∞. Then δ 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.

The image of Jordan left derivations on algebras

“…In the same paper he conjectured that every Jordan left derivation on a Banach algebra maps the algebra into its radical. Furthermore, in [11] the authors obtained a result as follows: If d is a Jordan left derivation on a unital Banach algebra A with the condition sup{r(c −1 d(c)) : c ∈ A invertible} < ∞, then d(A) ⊆ rad(A), where r(a) denotes the spectral radius of a ∈ A.…”