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Derivations, local derivations and atomic boolean subspace lattices

Published online by Cambridge University Press:  17 April 2009

Pengtong Li  and
Jipu Ma
Pengtong Li
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, People's Republic of China e-mail: [email protected], [email protected]
Jipu Ma
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, People's Republic of China
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Let ℒ be an atomic Boolean subspace lattice on a Banach space X. In this paper, we prove that if ℳ is an ideal of Alg ℒ then every derivation δ from Alg ℒ into ℳ is necessarily quasi-spatial, that is, there exists a densely defined closed linear operator T: 𝒟(T) ⊆ X → X with its domain 𝒟(T) invariant under every element of Alg ℒ, such that δ(A) x = (TA – AT) x for every A ∈ Alg ℒ and every x ∈ 𝒟(T). Also, if ℳ ⊆ ℬ(X) is an Alg ℒ-module then it is shown that every local derivation from Alg ℒ into ℳ is necessary a derivation. In particular, every local derivation from Alg ℒ into ℬ(X) is a derivation and every local derivation from Alg ℒ into itself is a quasi-spatial derivation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 3 , December 2002 , pp. 477 - 486
Copyright
Copyright © Australian Mathematical Society 2002

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