Complete distributivity and ordered group lattices

Laurie, Cecelia

doi:10.1090/S0002-9939-1985-0796450-6

Complete distributivity and ordered group lattices
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by Cecelia Laurie PDF

Proc. Amer. Math. Soc. 95 (1985), 79-82 Request permission

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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