Abstract
For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ayupov, S., Kudaybergenov, K.: 2-Local derivations on von Neumann algebras. Positivity 19, 445–455 (2014)
Ayupov, S., Kudaybergenov, K., Peralta, A.: A survey on local and 2-local derivations on C*- and von Neuman algebras. Top. Funct. Anal. Algebra Contemp. Math. 672, 73–126 (2016)
Christensen, E.: Derivations of nest algebras. Math. Ann. 229, 155–161 (1977)
Crist, R.: Local derivations on operator algebras. J. Funct. Anal. 135, 72–92 (1996)
Cusack, J.: Jordan derivations on rings. Proc. Am. Math. Soc. 53, 321–324 (1975)
Hadwin, D., Li, J.: Local derivations and local automorphisms on some algebras. J. Oper. Theory 60, 29–44 (2008)
He, J., Li, J., An, G., Huang, W.: Characterizations of 2-local derivations and local Lie derivations on some algebras. Sib. Math. J. (to appear) (2017)
Johnson, B.: Local derivations on C*-algebras are derivations. Trans. Am. Math. Soc. 353, 313–325 (2001)
Kadison, R.: Derivations of operator algebras. Ann. Math. 83, 280–293 (1966)
Kadison, R.: Local derivations. J. Algebra 130, 494–509 (1990)
Kim, S., Kim, J.: Local automorphisms and derivations on \(M_n({\mathbb{C}})\). Proc. Am. Math. Soc. 132, 1389–1392 (2004)
Lance, E.: Hilbert C*-Modules: A Toolkit for Operator Algebraists. Cambridge University Press, Cambridge (1995)
Larson, D., Sourour, A.: Local derivations and local automorphisms. Proc. Symp. Pure Math. 51, 187–194 (1990)
Li, J., Pan, Z.: Annihilator-preserving maps, multipliers and local derivations. Linear Algebra Appl. 432, 5–13 (2010)
Li, P., Han, D., Tang, W.: Derivations on the algebra of operators in Hilbert C*-modules. Acta Math. Sin. (Engl. Ser.) 28, 1615–1622 (2012)
Moghadam, M., Miri, M., Janfada, A.: A note on derivations on the algebra of operators in Hilbert C*-modules. Mediterr. J. Math. 13, 1167–1175 (2016)
Sakai, S.: Derivations of W*-algebras. Ann. Math. 83, 273–279 (1966)
Šemrl, P.: Local automorphisms and derivations on \(B(H)\). Proc. Am. Math. Soc. 125, 2677–2680 (1997)
Zhang, J., Li, H.: 2-Loacl derivations on digraph algebras. Acta Math. Sin. (Chin. Ser.) 49, 1401–1406 (2006)
Acknowledgements
This paper was partially supported by National Natural Science Foundation of China (Grant No. 11371136).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, J., Li, J. & Zhao, D. Derivations, Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules. Mediterr. J. Math. 14, 230 (2017). https://doi.org/10.1007/s00009-017-1032-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-1032-5