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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero

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Volume 49, Issue 3


Derivable Maps and Generalized Derivations on Nest and Standard Algebras

Zhidong Pan
Published Online: 2016-08-20 | DOI: https://doi.org/10.1515/dema-2016-0028


For an algebra A, an A-bimodule M, and m ∈ M, define a relation on A by RA(m,0)={(a, b) ∈A×A: amb =0}. We show that generalized derivations on unital standard algebras on Banach spaces can be characterized precisely as derivable maps on these relations. More precisely, if A is a unital standard algebra on a Banach space X then Δ ∈ L(A, B, (X)) is a generalized derivation if and only if Δ is derivable on RA(M, 0), for some M ∈ B(X). We give an example to show this is not the case in general for nest algebras. On the other hand, for an idempotent P in a nest algebra A = algN on a Hilbert space H such that P is either left-faithful to N or right-faithful to N, if δ ∈ L(A, B(H)) is derivable on RA(P, 0) then Δ is a generalized derivation.

Keywords: derivable map; derivation; nest algebra


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About the article

Received: 2015-01-13

Revised: 2015-04-20

Published Online: 2016-08-20

Published in Print: 2016-09-01

Citation Information: Demonstratio Mathematica, Volume 49, Issue 3, Pages 331–344, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.1515/dema-2016-0028.

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© by Zhidong Pan. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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