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

Published electronically:
December 31, 2003

MathSciNet review:
2053972

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a subspace lattice on a normed space containing a nontrivial comparable element. If commutes with all the operators in , then there exists a scalar such that . Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type , Type and Type , respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type , and that nontrivial atomic Boolean subspace lattices are Type .

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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