Lattice properties of subspace families in an inner product space (Q2718979)

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 \textit{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 \(S\).