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Proceedings of the American Mathematical Society

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Cyclic atoms in orthomodular lattices

Author: Donald E. Catlin
Journal: Proc. Amer. Math. Soc. 30 (1971), 412-418
MSC: Primary 06.40
MathSciNet review: 0285457
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Abstract: Let $ P(H)$ denote the projection lattice of a separable Hilbert space H. For each $ {\text{x}} \in H$, let $ {P_{\text{x}}}$ denote the projection onto the one dimensional subspace generated by x. If B is a Boolean sublattice of $ P(H)$, then it is a theorem that whenever B is maximal in $ P(H)$ there exists a vector $ {{\text{x}}_0} \in H$, called a cyclic vector for B, such that the join in $ P(H)$ of all the $ {P_{Q({{\text{x}}_0})}}$ as Q ranges through B is the identity operator I. In this paper we show that this theorem is an immediate corollary of a more general theorem in orthomodular lattice theory. In addition, a final theorem in the paper makes clear the necessity for the separability assumption on H.

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Keywords: Separable Hilbert space, cyclic vector, orthomodular lattice, Sasaki projection, atomic lattice, complete lattice, atomic bisection property, atomic projection property, block, cyclic atom, hyperoctant property
Article copyright: © Copyright 1971 American Mathematical Society

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