On complete-cocomplete subspaces of an inner product space

Abstract

In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space S is complete if and only if there exists a σ-additive state on C(S), the orthomodular poset of complete-cocomplete subspaces of S. We then consider the problem of whether every state on E(S), the class of splitting subspaces of S, can be extended to a Hilbertian state on E(\(\bar S\)); we show that for the dense hyperplane S (of a separable Hilbert space) constructed by P. Ptak and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on E(S) is a restriction of a state on E(\(\bar S\)).