Derivations, local derivations and atomic boolean subspace lattices
2002 ◽
Vol 66
(3)
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pp. 477-486
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Keyword(s):
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.
1984 ◽
Vol 27
(2)
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pp. 229-233
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1965 ◽
Vol 17
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pp. 1030-1040
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Keyword(s):
1968 ◽
Vol 8
(1)
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pp. 119-127
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2004 ◽
Vol 77
(1)
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pp. 73-90
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1969 ◽
Vol 12
(5)
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pp. 639-643
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1998 ◽
Vol 58
(2)
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pp. 245-260
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