Abstract
Let \({\mathcal {X}}\) be a Banach algebra, let \(\phi ,\psi \) be mappings on \({\mathcal {X}}\), let \(\delta \) be a \((\phi ,\psi )\)-derivation on \({\mathcal {X}}\) and let d be a generalized \((\phi ,\psi )\)-derivation related to \(\delta \). If \({\mathcal {X}}\) is simple, we determine some sufficient conditions under which every generalized \((\phi ,\psi )\)-derivation on \({\mathcal {X}}\) is continuous (without continuity of \(\delta \)). In addition, we show that if d is inner on \(\mathcal {F}_1(X)\) (the set of all rank one operators on \({\mathcal {X}})\) and \(\phi ,\psi : \mathcal {B}({\mathcal {X}})\longrightarrow \mathcal {B}({\mathcal {X}})\) are homomorphisms and surjective on \(\mathcal {F}_1(X)\) then d is inner on \(\mathcal {B}({\mathcal {X}})\). Finally, we characterize the linear mappings on \(\mathcal {B}({\mathcal {X}})\) which behave like generalized \((\phi ,\psi )\)-derivations when acting on zero products.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Christensen, E.: Derivations of nest algebras. Math. Ann. 299, 155–161 (1977)
Dales, H.D.: Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, vol. 24. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (2000)
Hejazian, S., Janfada, A.R., Mirzavaziri, M., Moslehian, M.S.: Achivement of continuity of \((\phi,\psi )\)-derivations. Bull. Belg. Math. Soc. Simon Stevn 14, 641–652 (2007)
Hou, C., Ming, Q.: Continuity of \((\alpha,\beta )\)-derivations of operator algebras. J. Korean Math. Soc. 48, 823–835 (2011)
Jing, W., Lu, S., Li, P.: Characterizations of derivations on some operators algebras. Bull. Austral Math. Soc. 66, 227–232 (2002)
Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Am. J. Math. 90, 1067–1073 (1968)
Kadison, R.V.: Derivations of operator algebras. Ann. Math. 83(2), 280–293 (1966)
Kaplansky, L.: Modules over operator algebras. Am. J. Math. 75, 839–858 (1953)
Mirzavaziri, M., Moslehian, M.S.: Automatic continuity of \(\sigma \)-derivations in \(C\)*-algebras. Proc. Am. Math. Soc. 134, 3319–3327 (2006)
Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. Lond. Math. Soc. 5(2), 432–438 (1972)
Sakai, S.: On a conjecture of Kaplansky. Tohoku Math. J. 12(2), 31–53 (1960)
Sakai, S.: Derivations of \(W^*\)-algebras. Ann. Math. 83(2), 280–293 (1966)
Villena, A.R.: Automatic continuity in associative and nonassociative context. Irish Math. Soc. Bull. 46, 43–76 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Janfada, A.R., Kafimoghadam, M. & Miri, M. Continuity and Structure of Generalized \(\varvec{(\phi ,\psi )}\)-Derivations. Results Math 72, 1813–1821 (2017). https://doi.org/10.1007/s00025-017-0731-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0731-3
Keywords
- Generalized \((\phi , \psi )\)-derivations
- automatic continuity
- innerness
- Banach algebras
- operators algebra