, Volume 64, Issue 1, pp 99–110 | Cite as

Characterization of some derivations on von Neumann algebras via left centralizers

  • A. Hosseini


Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\), where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \). By applying this conclusion, we offer a characterization of left \(\sigma \)-centralizers, generalized derivations and generalized \(\sigma \)-derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi-\(\sigma \)-derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.


Left (right) centralizer Semi Hochschild 2-cocycle Bi-\(\sigma \)-derivaton \(\mathcal {C}\)-derivation von Neumann algebra 

Mathematics Subject Classification

Primary 47B47 Secondary 17B40 46L10 



The author is greatly indebted to the referee for his/her valuable suggestions and careful reading of the paper.


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© Università degli Studi di Ferrara 2017

Authors and Affiliations

  1. 1.Department of MathematicsKashmar higher education InstituteKashmarIran

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