Gleason's theorem and completeness of inner product spaces
Let V be an inner product space over the field of real or complex numbers and let \({\mathcal L}(V)\) be the lattice of all strongly closed subspaces of V. It is shown that if the dimension of V is an infinite nonmeasurable cardinal and \({\mathcal L}(V)\) possesses at least one state, or V be an infinite-dimensional space and \({\mathcal L}(V)\) possesses at least one completely additive state, then V is complete. These assertions generalize the result of \textit{J. Hamhalter} and \textit{P. Ptak} [Bull. Lond. Math. Soc. 19, 259-263 (1987; Zbl 0601.46027)] which was established by them for a separable inner product space. It is addduced a version of Gleason's theorem for \(\sigma\)-finite measures on the lattice \({\mathcal L}(V)\) of a separable inner product space V. The analogous result for Hilbert spaces was obtained by \textit{G. D. Lugovaja} and \textit{A. N. Sherstnev} [Sov. Math. 24, No.12, 35-38 (1980; Zbl 0483.46042)].
lattice of all strongly closed subspaces
scientific article; zbMATH DE number 4084366
scientific article