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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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  • [ALVS] A. Loginov and V. Sulman, Hereditary and intermediate reflexivity of $ {W^ \ast }$-algebras, Izv. Akad. Nauk USSR 39 (1975), 1260-1273; English transl., Math. USSR-Izv. 9 (1975), 1189-1201. MR 0405124 (53:8919)
  • [DH] D. Hadwin, Algebraically reflexive linear transformations, Linear and Multilinear Algebra 14 (1983), 225-233. MR 718951 (85e:47003)
  • [H1] Han Deguang, Derivations and isomorphisms of certain reflexive operator algebras, submitted.
  • [H2] -, Rank one operators and bimodules of reflexive operator algebras in Banach spaces, J. Math. Anal. Appl. 161 (1991), 188-193. MR 1127556 (92h:47062)
  • [H3] -, Cohomology of certain reflexive operator algebras, submitted.
  • [H4] -, The first cohomology groups of nest algebras on normed spaces, Proc. Amer. Math. Soc. 118 (1993), 1147-1149. MR 1139465 (93j:47063)
  • [K] R. V. Kadison, Local derivations, J. Algebra 130 (1990), 494-509. MR 1051316 (91f:46092)
  • [L] D. R. Larson, Reflexivity, algebraic reflexivity and linear interpolation, Amer. J. Math. 110 (1988), 283-291. MR 935008 (89d:47096)
  • [LS] D. R. Larson and A. R. Sourour, Local derivations and local automorphisms of $ B(X)$, Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1992, pp. 187-192. MR 1077437 (91k:47106)
  • [Lam] M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, J. London Math. Soc. (2) 25 (1982), 499-512. MR 657507 (84f:47053)
  • [Spa] N. K. Spanoudakis, Generalizations of certain nest algebra results, Proc. Amer. Math. Soc. 115 (1992), 711-723. MR 1097353 (92i:47049)

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

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