Gleason's theorem and completeness of inner product spaces
type: Item
instance of: scholarly article
title: Gleason's theorem and completeness of inner product spaces
zbMATH Open document ID: 0663.46060
DOI: 10.1007/BF00669390
author: Ladislav Mišík
author: Anatolij Dvurečenskij
published in: International Journal of Theoretical Physics
publication date: 1988-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 4084366
zbMATH Keywords: inner product space
zbMATH Keywords: lattice of all strongly closed subspaces
zbMATH Keywords: completely additive state
zbMATH Keywords: Gleason's theorem
reviewed by: Vladimir Chilin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4084366