Derivable Maps and Generalized Derivations on Nest and Standard Algebras
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 alge...
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| Format: | Article |
| Language: | English |
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De Gruyter
2016-09-01
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| Series: | Demonstratio Mathematica |
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| Online Access: | http://www.degruyter.com/view/j/dema.2016.49.issue-3/dema-2016-0028/dema-2016-0028.xml?format=INT |
| Summary: | 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. |
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| ISSN: | 0420-1213 2391-4661 |