Range of Charges on Orthogonally Closed Subspaces of an Inner Product Space

It is still an open question whether the complete lattice F(S) of all orthogonally closed subspaces of an incomplete inner product space S admits a nonzero charge. A negative answer would result in a new way of completeness characterization of inner product spaces. Many partial results have been established regarding what has now turned to be a highly nontrivial problem. Recently, in Dvurečenskij and Ptak (Letters in Mathematical Physics, 62, 63–70, 2002) the range of a finitely additive state s on F(S), dim S = ∞, was shown to include the whole interval [0, 1]. This was then generalized in Dvurečenskij (International Journal of Theoretical Physics, 2003) for general inner product spaces satisfying the Gleason property. Motivated by these results, we give a thorough investigation of the possible ranges of charges on F(S), dim S ≥q3. We show that if the nonzero charge m is bounded, then for infinite dimensional inner product spaces, Range(m) is always convex. We also show that this need not be the case with unbounded charges. Finally, in the last section, we investigate the range of charges on F(S), dim S = ∞, satisfying the sign-preserving and Jauch-Piron properties. We show that for such measures the range is again always convex.