Lattice properties of subspace families in an inner product space
Pavel Pták, Hans Weber · Proceedings of the American Mathematical Society · 2001
Let S S be a separable inner product space over the field of real numbers. Let E ( S ) E(S) (resp., C ( S ) ) C(S)) denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of S S . We ask whether E ( S ) E(S) (resp., C ( S ) ) C(S)) can be a lattice without S S being complete (i.e. without S 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 S that E ( S ) E(S) is not a lattice and C ( S ) C(S) is a (modular) lattice. We then go on showing that the orthomodular poset E ( S ) E(S) may not be a lattice even if E ( S ) = C ( S ) E(S)=C(S) . Finally, we construct a noncomplete space S