Abstract
The main purpose of this paper is to prove that every generalized derivation on a commutative von Neumann algebra is identically zero.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brešar, M.: On the distance of compositions of two derivations of the generalized derivations. Glasg. Math. J. 33, 89–93 (1991)
Bratteli, O., Robinson, D.W.: Banach Algebras and Automatic Continuity. \(C^{\ast }\)- and \(W^{\ast }\)-algebras, symmetry groups, decomposition of states, Second edition. Texts and Monographs in Physics. Springer, New York (1987)
Bratteli, O., Robinson, D.W.: Banach Algebras and Automatic Continuity. Equilibrium states. Models in quantum statistical mechanics Second edition. Texts and Monographs in Physics. Springer, Berlin (1997)
Dales, H.G.: Banach Algebras and Automatic Continuity. London Math. Soc. Monographs, New Series, 24. Oxford University Press, New York (2000)
Dales, H.G., Aiena, Pietro, Eschmeier, Jorg, Laursen, Kjeld, Willis, George A.: Introduction to Banach Algebras, Operators and Harmonic Analysis. Cambridge University Press, Cambridge (2003)
Hosseini, A., Hassani, M., Niknam, A.: On the Range of a Derivation. Iran. J. Sci. Technol. Trans. A Sci. 38A2, 111–115 (2014)
Hosseini, A.: A characterization of derivations on uniformly mean value Banach algebras. Turk. J. Math. 40, 1058–1070 (2016)
Hosseini, A.: Some conditions under which derivations are zero on Banach \(\ast \)-algebras, Demonstr. Math. (to appear in issue 3 or 4 (50) (2017)
Jung, Y.S.: On the Invariance of Primitive Ideals via \(\phi \)-derivations on Ba- nach Algebras. Kyungpook Math. J. 53, 497–505 (2013)
Kaplansky, I.: Projections in Banach Algebras. Ann. Math. 53(2), 235–249 (1951)
Kim, G.H.: A result concerning derivations in noncommutative Banach algebras. Sci. Math. Jpn. 4, 193–196 (2001)
Mirzavaziri, M., Omidvar Tehrani, E.: Generalized higher derivations are sequences of generalized derivations. J. Adv. Res. Pure Math. 3, 75–86 (2011)
Mirzavaziri, M.: Characterization of higher derivations on algebras. Commun. Algebra 38(3), 981–987 (2010)
Mathieu, M.: Where to find the image of a derivation. Banach Center Publ. 30, 237–249 (1994)
Murphy, G.J.: \(C^{\ast }\)-Algebras and Operator Theory. Academic Press, Boston (1990)
Shahzad, N.: Jordan and left derivations on locally \(C^{\ast }\)-algebras. Southeast Asian Bull. Math. 31, 1183–1190 (2007)
Singh, B.: Higher derivations. Normal flatness, and analytic products. J. Algebra 95, 236–244 (1985)
Sait\(\widehat{o}\), K., Maitland Wright, J.D.: On Defining AW*-algebras and Rickart C*-algebras. arXiv:1501.02434v1 [math. OA] 11 Jan 2015
Singer, I.M., Wermer, J.: Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955)
Sakai, S.: On a conjecture of Kaplansky. Tôhoku Math. J. 12, 31–33 (1960)
Thomas, M.P.: The image of a derivation is contained in the radical. Ann. Math. 128, 435–460 (1988)
Thomas, M.P.: Primitive ideals and derivations on non-commutative Banach algebras. Pac. J. Math. 159, 139–152 (1993)
Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (2001)
Acknowledgements
The author is greatly indebted to the referee for his/her valuable suggestions and careful reading of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Madjid Mirzavaziri.
Rights and permissions
About this article
Cite this article
Hosseini, A. Generalized derivations on commutative Von Neumann algebras. Rend. Circ. Mat. Palermo, II. Ser 67, 1–6 (2018). https://doi.org/10.1007/s12215-016-0283-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-016-0283-5