Commutants of reflexive algebras and classification of completely distributive subspace lattices |
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| Authors: | Pengtong Li Shijie Lu Jipu Ma |
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| Affiliation: | Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of China ; Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China ; Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China |
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| 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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| Keywords: | Reflexive algebras commutants complete distributivity comparable elements rank one operators |
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