Invariant linear manifolds for CSL-algebras and nest algebras

Abstract

Every invariant linear manifold for a CSL-algebra, Alg ⁡ L \operatorname {Alg} \mathcal {L} , is a closed subspace if, and only if, each non-zero projection in L \mathcal {L} is generated by finitely many atoms associated with the projection lattice. When L \mathcal {L} is a nest, this condition is equivalent to the condition that every non-zero projection in L \mathcal {L} has an immediate predecessor ( L ⊥ \mathcal {L}^{\perp } is well ordered). The invariant linear manifolds of a nest algebra are totally ordered by inclusion if, and only if, every non-zero projection in the nest has an immediate predecessor.