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

Document Type : Research Paper

Author

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

Abstract

Singer and Wermer proved that if A is a commutative Banach algebra and d:AA is a continuous derivation, then d(A)rad(A), where rad(A) denotes the Jacobson radical of 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

Volume 11, Issue 1
April 2020
Pages 453-471
  • Receive Date: 10 February 2019
  • Revise Date: 12 March 2019
  • Accept Date: 17 March 2019