2024/03/27 by Amin Hosseini, Hosseini, Amin, Antonio M. Peralta +3 · 2 citations
Computer Science · Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2403.18773
openalex publication_date 2024/03/27 · openalex created_date 2024/03/29 · openalex updated_date 2026/07/28
Let M be a Banach bimodule over an associative Banach algebra A, and let F: A→ M be a linear mapping. Three main uses of the term generalized derivation are identified in the available literature, namely, (\checkmark) F is a generalized derivation of the first type if there exists a derivation d : A→ M** satisfying F(a b ) = F(a) b + a d(b), for all a,b∈ A. (\checkmark) F is a generalized derivation of the second type if there exists an element ξ∈ M** satisfying F(a b ) = F(a) b + a F(b) - a ξb, for all a,b∈ A. (\checkmark) F is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings G,H : A→ M satisfying F(a b ) = G(a) b + a H(b), for all a,b∈ A. These three types of maps are not, in general, equivalent. Although the first two notions are well studied when A is a C^*-algebra, their connections with the third one have not yet been explored. In this note we prove that every generalized derivation of the third type from a C^*-algebra A to a Banach A-bimodule M is automatically continuous. We also show that every (continuous) generalized derivation of the third type from A to M is a generalized derivation of the first and second type. Consequently, the three notions coincide in this case. We also explore some concepts of generalized Jordan derivations on a C^*-algebra and establish some continuity properties for them.