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Lattice properties of subspace families in an inner product space

Authors: Pavel Pták and Hans Weber
Journal: Proc. Amer. Math. Soc. 129 (2001), 2111-2117
MSC (2000): Primary 03G12, 46C05, 81P10
Published electronically: February 15, 2001
MathSciNet review: 1825924
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Let $S$ be a separable inner product space over the field of real numbers. Let $E(S)$ (resp., $C(S))$ denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of $S$. We ask whether $E(S)$ (resp., $C(S))$ can be a lattice without $S$ being complete (i.e. without $S$ being Hilbert). This question is relevant to the recent study of the algebraic properties of splitting subspaces and to the search for ``nonstandard'' orthomodular spaces as motivated by quantum theories. We first exhibit such a space $S$ that $E(S)$ is not a lattice and $C(S)$ is a (modular) lattice. We then go on showing that the orthomodular poset $E(S)$ may not be a lattice even if $E(S)=C(S)$. Finally, we construct a noncomplete space $S$ such that $E(S)=C(S)$ with $E(S)$ being a (modular) lattice. (Thus, the lattice properties of $E(S)$ (resp. $C(S))$ do not seem to have an explicit relation to the completeness of $S$ though the Ammemia-Araki theorem may suggest the opposite.) As a by-product of our construction we find that there is a noncomplete $S$ such that all states on $E(S)$ are restrictions of the states on $E(\overline{S})$ for $\overline{S}$ being the completion of $S$ (this provides a solution to a recently formulated problem).

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Additional Information

Pavel Pták
Affiliation: Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University, 166 27 Prague 6, Czech Republic

Hans Weber
Affiliation: Dipartimento di Matematica, e Informatica, Università degli Studi di Udine, I-33100 Udine, Italy

Keywords: Inner product (= prehilbert) space, splitting subspace, orthomodular poset, lattice.
Received by editor(s): September 30, 1998
Received by editor(s) in revised form: June 2, 1999
Published electronically: February 15, 2001
Additional Notes: The authors acknowledge the support of grant GAČR 201/98/1153 of the Czech Grant Agency and Progetto di ricerca di interesse nazionale Analisi Reale (Italy).
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society