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Commutants of reflexive algebras and classification of completely distributive subspace lattices


Authors: Pengtong Li, Shijie Lu and Jipu Ma
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2005-2012
MSC (2000): Primary 47L35, 47L75
DOI: https://doi.org/10.1090/S0002-9939-03-07325-8
Published electronically: December 31, 2003
MathSciNet review: 2053972
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Abstract: Let $\mathcal{L}$ be a subspace lattice on a normed space $X$ containing a nontrivial comparable element. If $T$ commutes with all the operators in $\mbox{Alg}\mathcal{L}$, then there exists a scalar $\lambda$ such that $(T-\lambda I)^2=0$. Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type $I^{(n)}$, Type $II^{(n)}$ and Type $III$, respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type $I^{(1)}$, and that nontrivial atomic Boolean subspace lattices are Type $II^{(0)}$.


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Additional Information

Pengtong Li
Affiliation: Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China
Email: pengtonglee@vip.sina.com

Shijie Lu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Address at time of publication: City College, Zhejiang University, Hangzhou 310015, People’s Republic of China
Email: lusj@zucc.edu.cn

Jipu Ma
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-03-07325-8
Keywords: Reflexive algebras, commutants, complete distributivity, comparable elements, rank one operators
Received by editor(s): November 19, 2001
Received by editor(s) in revised form: March 16, 2003
Published electronically: December 31, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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