Cyclic atoms in orthomodular lattices

Author:
Donald E. Catlin

Journal:
Proc. Amer. Math. Soc. **30** (1971), 412-418

MSC:
Primary 06.40

DOI:
https://doi.org/10.1090/S0002-9939-1971-0285457-3

MathSciNet review:
0285457

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the projection lattice of a separable Hilbert space *H*. For each , let denote the projection onto the one dimensional subspace generated by *x*. If *B* is a Boolean sublattice of , then it is a theorem that whenever *B* is maximal in there exists a vector , called a *cyclic vector for B*, such that the join in of all the 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0285457-3

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