States on families of subspaces of pre-Hilbert spaces
type: Item
instance of: scholarly article
title: States on families of subspaces of pre-Hilbert spaces
zbMATH Open document ID: 0696.46023
DOI: 10.1007/BF00420009
published in: Letters in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: Considering S a complex pre-Hilbert space, the author defines the following six families of closed subspaces of this one. (1) W(S) the set of all subspaces of S; (2) F(S) the set of all subspaces M of S for which we have \(M=M^{\perp \perp};\) (3) D(S) the set of all subspaces M of S for which there exists an orthonormal system (ONS) \(\{u_ i\}\) such that \(M=\{u_ i\}^{\perp \perp};\) (4) R(S) the set of all subspaces M of S such that \(M=\{u_ i\}^{\perp \perp}\) for all maximal ONS (MONS for short); (5) V(S) the set of all subspaces M of S such that \(M=\{u_ i\}^{\perp \perp}\) and \(M^{\perp}=\{v_ j\}^{\perp \perp}\) for every MONS \(\{u_ i\}\) and \(\{v_ j\}\) in M and \(M^{\perp}\), respectively. (6) E(S) the set of all subspaces M of S for which \(M+M^{\perp}=S.\) One may prove that the following is true: \[ E(S)\subseteq V(S)\subseteq R(S)\subseteq D(S)\subseteq F(S)\subseteq W(S) \] and E(S) contains any complete and therefore any finite-dimensional subspace. One may prove too, that if S is a Hilbert space, then all six families are identical. In this work, the author proves that if an arbitrary one of the six families of subspaces possesses at least one completely additive state, then the pre-Hilbert space is complete. These results play a considerable role in the axiomatic model of quantum mechanics.
reviewed by: A.Donescu
Mathematics Subject Classification ID: 46C99
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 4140532
zbMATH Keywords: completely additive state
zbMATH Keywords: axiomatic model of quantum mechanics
author: Anatolij Dvurečenskij
MaRDI profile type: Publication
cites work: Q3778488
cites work: Completeness of inner product spaces with respect to splitting subspaces
cites work: Completeness of inner product spaces and quantum logic of splitting subspaces
cites work: Gleason's theorem and completeness of inner product spaces
cites work: State on splitting subspaces and completeness of inner product spaces
cites work: Inner Product Spaces
cites work: A Completeness Criterion for Inner Product Spaces
full work available at URL: https://doi.org/10.1007/bf00420009
OpenAlex ID: W1989155588
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description: scientific article; zbMATH DE number 4140532