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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Complete distributivity and ordered group lattices


Author: Cecelia Laurie
Journal: Proc. Amer. Math. Soc. 95 (1985), 79-82
MSC: Primary 47D25; Secondary 47A15
MathSciNet review: 796450
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Abstract: Arveson has recently generalized an important result of Andersen's about continuous nests to a larger class of lattices. Andersen's result is a base for much of the recent interesting work on compact perturbations and similarity of nest algebras. This paper investigates further the structure of such lattices. It is shown that the ordered group lattices with $ \Sigma $-continuous measures introduced by Arveson are completely distributive. This immediately implies various nice properties of $ {\text{Alg}}\mathcal{L}$, the associated algebra of operators leaving such a lattice $ \mathcal{L}$ invariant. (Among these are the fact that the rank one operators in $ {\text{Alg}}\mathcal{L}$ are dense in $ {\text{Alg}}\mathcal{L}$ and that $ {\text{Alg}}\mathcal{L} + \mathcal{K}$ is norm closed where $ \mathcal{K}$ denotes the compact operators.)


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0796450-6
PII: S 0002-9939(1985)0796450-6
Article copyright: © Copyright 1985 American Mathematical Society