Commutants of reflexive algebras and classification of completely distributive subspace lattices

Li, Pengtong; Lu, Shijie; Ma, Jipu

doi:10.1090/S0002-9939-03-07325-8

Commutants of reflexive algebras and classification of completely distributive subspace lattices
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by Pengtong Li, Shijie Lu and Jipu Ma PDF

Proc. Amer. Math. Soc. 132 (2004), 2005-2012 Request permission

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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