Order properties of splitting subspaces in an inner product space

Year, pages: 2004, 2111 - 2117

Let $E(S)$ (resp. $C(S)$) be the orthomodular poset of all splitting subspaces (resp. all complete-cocomplete subspaces) in an inner product space $S$. As is known, neither $E(S)$ nor $C(S)$ has to be a lattice ([PTÁK, P.—WEBER, H.: Lattice properties of subspace families in an inner product space, Proc. Amer. Math. Soc. 129 (2001), 2111–2117]). In this note we test $E(S)$ (resp. $C(S)$) for order properties which are ``lattice-like''. We show that, in general, either $E(S)$ or $C(S)$ does not have to enjoy the Riesz Interpolation Property. On the other hand, both $E(S)$ and $C(S)$ do possess the regularity property as dealt with in quantum logics (see [HARDING, J.: Regularity in quantum logics, Internat. J. Theoret. Phys. 37 (1998), 1173–1212]). In the final observation, we show that a very weak form of countable lattice completeness implies the (topological) completeness of $S$, contributing slightly to the investigations carried on in [DVUREČENSKIJ, A.: Gleason's Theorem and Applications, Kluwer Acad. Publ., Dordrecht-Boston-London, 1993], and elsewhere (the lattice property of $E(S)$ or $C(S)$ is known to be too weak to imply completeness of $S$, see [PTÁK, P.—WEBER, H.: Lattice properties of subspace families in an inner product space, Proc. Amer. Math. Soc. 129 (2001), 2111–2117]).

Pták, P., Weber, H. 2004. Order properties of splitting subspaces in an inner product space. In Mathematica Slovaca, vol. 54, no.2, pp. 2111-2117. 0139-9918.

Pták, P., Weber, H. (2004). Order properties of splitting subspaces in an inner product space. Mathematica Slovaca, 54(2), 2111-2117. 0139-9918.