Derivable Maps and Generalized Derivations on Nest and Standard Algebras

Demonstratio mathematica

article

Zhidong Pan¹ 

[1] SAGINAW VALLEY STATE UNIVERSITY, UNIVERSITY CENTER

关键词: derivable map;    derivation;    nest algebra;   

DOI  :  10.1515/dema-2016-0028

学科分类：外科医学

来源: De Gruyter

【 摘 要 】

For an algebra A, an A-bimodule M, and m ∈ M, define a relation on A by R A (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 R A (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 R A (P, 0) then Δ is a generalized derivation.

【 授权许可】

CC BY