A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.

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.

Keywords:

Derivation , Jordan derivation , Singer-Wermer Theorem , {g , h}-derivation , {g , h}-homomorphism