Commutants of reflexive algebras and classification of completely distributive subspace lattices
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- by Pengtong Li, Shijie Lu and Jipu Ma PDF
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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 $II^{(0)}$.References
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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
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2053972