Singer and Wermer proved that if $\mathcal{A}$ is a commutative Banach algebra and $d: \mathcal{A}\longrightarrow \mathcal{A}$ is a continuous derivation, then $d(\mathcal{A}) ⊆ rad(\mathcal{A})$, where $rad(\mathcal{A})$ denotes the Jacobson radical of $\mathcal{A}$. In this paper, we establish a new proof of that theorem. Moreover, we prove that every continuous Jordan derivation on a finite dimensional Banach algebra, under certain conditions, is identically zero. As another objective of this article, we study {g, h}-derivations on algebras. In this regard, we prove that if f is a {g, h}-derivation on a unital algebra, then f, g and h are generalized derivations. Additionally, we achieve some results concerning the automatic continuity of {g, h}-derivations on Banach algebras. In the last section of the article, we introduce the concept of a {g, h}-homomorphism and then we present a characterization of it under certain conditions.
Hosseini,A . (2020). A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.. International Journal of Nonlinear Analysis and Applications, 11(1), 453-471. doi: 10.22075/ijnaa.2019.17189.1915
MLA
Hosseini,A . "A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.", International Journal of Nonlinear Analysis and Applications, 11, 1, 2020, 453-471. doi: 10.22075/ijnaa.2019.17189.1915
HARVARD
Hosseini A. (2020). 'A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.', International Journal of Nonlinear Analysis and Applications, 11(1), pp. 453-471. doi: 10.22075/ijnaa.2019.17189.1915
CHICAGO
A Hosseini, "A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.," International Journal of Nonlinear Analysis and Applications, 11 1 (2020): 453-471, doi: 10.22075/ijnaa.2019.17189.1915
VANCOUVER
Hosseini A. A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.. IJNAA. 2020;11(1):453-471. doi: 10.22075/ijnaa.2019.17189.1915
International Journal of Nonlinear Analysis and Applications