Lattice properties of subspace families in an inner product space

Abstract:

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).