Boletim da Sociedade Paranaense de MatemΓ‘tica (May 2024)
About the image of strongly generalized derivations of order $n$
Abstract
Let A and B be two algebras. A linear mapping π:A β B is called a strongly generalized derivation of order n, if there exist the families {πΈ_π: π΄ β π΅}_{π = 1}^{π}, {πΉ_π: π΄ β π΅}_{π = 1}^{π}, {πΊ_π: π΄ β π΅}_{π = 1}^{π} πππ {π»_π: π΄ β π΅}_{π = 1}^{π} of linear mappings which satisfy π₯(ππ) = Ξ£ [πΈ_π(π) πΉ_π(π) + πΊ_π(π)π»_π(π)] ππ =1 for all π, π β² π΄. The main purpose of this paper is to study the image of such derivations. Our main result on the image of strongly generalized derivations of order one reads as follows: Let A be a unital, commutative Banach algebra and let π₯: π΄ β π΄ be a continuous strongly generalized derivation of order one; that is, there exist the linear mappings πΈ, πΉ, πΊ, π»: π΄ β π΄ satisfying π·(ππ) = πΈ(π) πΉ(π) + πΊ(π) π»(π) for all π, π β² π΄. Let πΈ, πΉ, πΊ and π» be continuous linear mappings. We prove that, under certain conditions, π» (π΄), πΈ(π΄) πππ π₯(π΄) are contained in the Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image of continuous derivations on commutative Banach algebras.