Identities related to generalized derivations and Jordan (∗, ?)-derivations

Amin Hosseini

Filomat, Tome 35 (2021) no. 7, p. 2349

Résumé

The main purpose of this research is to characterize generalized (left) derivations and Jordan (∗, ?)-derivations on Banach algebras and rings using some functional identities. LetAbe a unital semiprime Banach algebra and let F,G : A→A be linear mappings satisfying F(x) = −x2G(x−1) for all x ∈ Inv(A), where Inv(A) denotes the set of all invertible elements ofA. Then both F and G are generalized derivations onA. Another result in this regard is as follows. LetA be a unital semiprime algebra and let n > 1 be an integer. Let f , 1 : A → A be linear mappings satisfying f (an) = nan−11(a) = n1(a)an−1 for all a ∈ A. If 1(e) ∈ Z(A), then f and 1 are generalized derivations associated with the same derivation onA. In particular, ifA is a unital semisimple Banach algebra, then both f and 1 are continuous linear mappings. Moreover, we define a (∗, ?)-ring and a Jordan (∗, ?)-derivation. A characterization of Jordan (∗, ?)-derivations is presented as follows. Let R be an n!-torsion free (∗, ?)-ring, let n > 1 be an integer and let d : R → R be an additive mapping satisfying d(an) = ∑n j=1 a? n− jd(a)a∗ j−1 for all a ∈ R. Then d is a Jordan (∗, ?)-derivation on R. Some other functional identities are also investigated

Détail

DOI : 10.2298/FIL2107349H

Classification : 47B47, 39B05
Keywords: generalized derivation, generalized left derivation, Jordan (∗, ?)-derivation, Banach algebra, (∗, ?)-ring