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Complete distributivity and ordered group lattices


Author: Cecelia Laurie
Journal: Proc. Amer. Math. Soc. 95 (1985), 79-82
MSC: Primary 47D25; Secondary 47A15
DOI: https://doi.org/10.1090/S0002-9939-1985-0796450-6
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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  • [1] N. T. Anderson, Compact perturbations of reflexive algebras, J. Funct. Anal. 38 (1980), 366-400. MR 593086 (82c:47005)
  • [2] -, Similarity of continuous nests, Bull. London Math. Soc. 15 (1983), 131-132. MR 689244 (85b:47049)
  • [3] W. Arveson, Perturbation theory for groups and lattices, J. Funct. Anal. 53 (1983), 22-73. MR 715546 (85d:47042)
  • [4] K. R. Davidson, Similarity and compact perturbations of nest algebras, J. Reine Angew. Math. 348 (1984), 72-87. MR 733923 (86c:47062)
  • [5] T. Fall, W. Arveson and P. Muhly, Perturbations of nest algebras, J. Operator Theory 1 (1979), 137-150. MR 526295 (80f:47035)
  • [6] A. Hopenwasser, C. Laurie and R. Moore, Reflexive algebras with completely distributive subspace lattices, J. Operator Theory 11 (1984), 91-108. MR 739795 (85m:47047)
  • [7] J. Kraus, Tensor products of reflexive algebras, J. London Math. Soc. (2) 28 (1983), 350-358. MR 713389 (85g:47063a)
  • [8] M. S. Lambrou, Completely distributive lattices, Fund. Math. 119 (1983), 227-240. MR 747026 (85g:06008)
  • [9] E. C. Lance, Cohomology and perturbations of nest algebras, Proc. London Math. Soc. (3) 43 (1981), 334-356. MR 628281 (83b:47053)
  • [10] D. R. Larson, Nest algebras and similarity transforms, Ann. of Math. (to appear). MR 794368 (86j:47061)
  • [11] C. Laurie and W. E. Longstaff, A note on rank-one operators in reflexive algebras.,Proc. Amer. Math. Soc. 89 (1983), 293-297. MR 712641 (85b:47052)
  • [12] W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), 491-498. MR 0394233 (52:15036)

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DOI: https://doi.org/10.1090/S0002-9939-1985-0796450-6
Article copyright: © Copyright 1985 American Mathematical Society

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