Document Type : Research Paper

Author

• Amin Hosseini

Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran

Abstract

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