Cyclic atoms in orthomodular lattices

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.