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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Local derivations of nest algebras

Authors: De Guang Han and Shu Yun Wei
Journal: Proc. Amer. Math. Soc. 123 (1995), 3095-3100
MSC: Primary 47D25; Secondary 47B47
MathSciNet review: 1246521
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Abstract: Let X be an arbitrary reflexive Banach space, and let $ \mathcal{N}$ be a nest on X. Denote by $ \mathcal{D}(\mathcal{N})$ the set of all derivations from $ \operatorname{Alg}\mathcal{N}$ into $ \operatorname{Alg}\mathcal{N}$. For $ N \subset \mathcal{N}$, we set $ {N_ - } = \vee \{ M \in \mathcal{N}:M \subset N\} $. We also write $ {0_ - } = 0$. Finally, for $ E, F \in \mathcal{N}$ define $ (E,F] = \{ K \in \mathcal{N}:E \subset K \subseteq F\} $. In this paper we prove that a sufficient condition for $ \mathcal{D}(\mathcal{N})$ to be (topologically) algebraically reflexive is that for all $ 0 \ne E \in \mathcal{N}$ and for all $ X \ne F \in \mathcal{N}$, there exist $ M \in (0,E]$ and $ N \in (F,X]$, such that $ {M_ - } \subset M$ and $ {N_ - } \subset N$. In particular, we prove that this condition automatically holds for nests acting on finite-dimensional Banach spaces.

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Keywords: Local derivation, nest algebras
Article copyright: © Copyright 1995 American Mathematical Society