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Lattice properties of subspace families in an inner product space. (English) Zbl 0968.03077

Let \(S\) be a separable inner product space over the reals. Let \(E(S)\) be the set of all subspaces \(A\) of \(S\) which are splitting, i.e., \(A+A^{\perp}=S\). Let \(C(S)\subseteq E(S)\) be the set of all closed subspaces of \(S\) and their (orthogonal) complements. Both \(E(S)\) and \(C(S)\) are orthomodular posets. The completeness of \(S\) may be equivalently formulated by numerous conditions on posets of subspaces of \(S\) [see A. Dvurecenskij, Gleason’s theorem and its applications. Kluwer, Dordrecht (1993; Zbl 0795.46045)] for an overview. Some open problems formulated there are solved in the paper under review. First, an example is given when \(E(S)\) is not a lattice, while \(C(S)\) is a modular ortholattice. Even if \(E(S)=C(S)\), it need not be a lattice. In contrast to this, \(E(S)=C(S)\) may be a modular ortholattice even if \(S\) is noncomplete. Thus the Amemiya-Araki theorem does not have an analogue for the posets \(E(S)\) and \(C(S)\). As a by-product, the authors construct a noncomplete space \(S\) such that all states on \(E(S)\) admit extensions to \(E(\overline{S})\), where \(\overline{S}\) is the completion of \(S\).

MSC:

03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Citations:

Zbl 0795.46045
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References:

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