Derivable Maps and Generalized Derivations on Nest and Standard Algebras

• Main Author: Pan Zhidong
• Format: Article
• Language: English
• Published: De Gruyter 2016-09-01
• Series: Demonstratio Mathematica
• Subjects: derivable map; derivation; nest algebra
• Online Access: http://www.degruyter.com/view/j/dema.2016.49.issue-3/dema-2016-0028/dema-2016-0028.xml?format=INT

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.