Generalized derivations on unital algebras determined by action on zero products

• Published in: Linear algebra and its applications Vol. 445; pp. 347 - 368
• Main Authors: Benkovič, Dominik, Grašič, Mateja
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.03.2014
• Subjects: Derivation; Generalized derivable mapping at zero point; Generalized derivation; Triangular algebra; Unital algebra; Zero product determined algebra; Generalized derivation; Derivation; 16W25; Generalized derivable mapping at zero point; Zero product determined algebra; Unital algebra; Triangular algebra
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2013.12.010

Let A be a unital algebra having a nontrivial idempotent and let M be a unitary A-bimodule. We consider linear maps F,G:A→M satisfying F(x)y+xG(y)=0 whenever x,y∈A are such that xy=0. For example, when A is zero product determined algebra (e.g. algebra generated by idempotents) F and G are generalized derivations F(x)=F(1)x+D(x) and G(x)=xG(1)+D(x) for all x∈A, where D:A→M is a derivation. If A is not generated by idempotents then there exist also nonstandard solutions for maps F and G. In the case of A being a triangular algebra under some condition on bimodule M the characterization of maps F and G is given. We also consider conditions on algebra A making it a zero product determined algebra.