Commutants of reflexive algebras and classification of completely distributive subspace lattices

Abstract

Let L \mathcal {L} be a subspace lattice on a normed space X X containing a nontrivial comparable element. If T T commutes with all the operators in Alg L \mbox {Alg}\mathcal {L} , then there exists a scalar λ \lambda such that ( T − λ I ) 2 = 0 (T-\lambda I)^2=0 . Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type I ( n ) I^{(n)} , Type I I ( n ) II^{(n)} and Type I I I III , respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type I ( 1 ) I^{(1)} , and that nontrivial atomic Boolean subspace lattices are Type I I ( 0 ) II^{(0)} .