Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

Buhagiar, David; Chetcuti, Emmanuel; Weber, Hendrik

doi:10.1016/j.jmaa.2008.04.024

Cited by 11 publications

(9 citation statements)

References 10 publications

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“…This was partly proved in Ref. 6, where it was shown that if S admits an orthonormal basis (i.e., a total orthonormal set; we recall that a subset X of S is total if spanX is dense in S) then E q (S) = E(S) if, and only if, S is complete. (In particular, this means that E(S) E q (S) when S is an incomplete separable pre-Hilbert space.)…”

Section: Introductionmentioning

confidence: 87%

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Quasi-splitting subspaces and Foulis-Randall subspaces

Buhagiar

Chetcuti

Dvurečenskij

2011

Journal of Mathematical Physics

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Nine theorems on the unification of quantum mechanics and relativityFor a pre-Hilbert space S, let F(S) denote the orthogonally closed subspaces, E q (S) the quasi-splitting subspaces, E(S) the splitting subspaces, D(S) the Foulis-Randall subspaces, and R(S) the maximal Foulis-Randall subspaces, of S. It was an open problem whether the equalities D(S) = F(S) and E(S) = R(S) hold in general [Cattaneo, G. and Marino, G., "Spectral decomposition of pre-Hilbert spaces as regard to suitable classes of normal closed operators," Boll. Unione Mat. Ital. 6 1-B, 451-466 (1982); Cattaneo, G., Franco, G., and Marino, G., "Ordering of families of subspaces of pre-Hilbert spaces and Dacey pre-Hilbert spaces," Boll. Unione Mat. Ital. 71-B, 167-183 (1987); Dvurečenskij, A., Gleason's Theorem and Its Applications (Kluwer, Dordrecht, 1992), p. 243.]. We prove that the first equality is true and exhibit a pre-Hilbert space S for which the second equality fails. In addition, we characterize complete pre-Hilbert spaces as follows: S is a Hilbert space if, and only if, S has an orthonormal basis and E q (S) admits a non-free charge. C

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Section: Introductionmentioning

confidence: 87%

“…The question of when does a pre-Hilbert space admit an orthonormal basis was also studied in Ref. 6.…”

Section: Introductionmentioning

confidence: 99%

Quasi-splitting subspaces and Foulis-Randall subspaces

Buhagiar

Chetcuti

Dvurečenskij

2011

Journal of Mathematical Physics

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“…The members of E q (S) should behave somewhat similarly to the splitting subspaces but on the other hand, E q (S) (at least when S is separable) is always furnished with a "good supply" of infinite Boolean σ -subalgebras. In [6] it was proved that S is complete if, and only if, S admits an orthonormal basis and E(S) = E q (S). In particular, when S is separable, this result gives a proper strengthening of the Amemiya-Araki Theorem.…”

Section: C(s) E(s) E Q (S) F(s)mentioning

confidence: 99%

Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Buhagiar

Chetcuti

Weber

2014

Mathematische Nachrichten

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Key words Pre-Hilbert space (=inner product space), orthogonally-closed subspace, quasi-splitting subspace, splitting subspace, riesz interpolation property MSC (2010) 46C15, 81P10Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces C(S), the splitting subspaces E(S), the quasi-splitting subspaces E q (S) and the orthogonally-closed subspaces F(S). It is well-known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion S i.e. d( S/S) = 1 then C(S) = E(S) and E q (S) = F(S). On the other extreme, when S = c 00 i.e. d( S/S) = 2 ℵ 0 then C(S) = E(S) and E q (S) = F(S). Motivated by thisand in contrast to it, we show that in general the codimension of S in S bears very little relation to the properties of these families. In particular, we show that the equalities E(S) = C(S) and E q (S) = F(S) can hold for inner product spaces with arbitrary codimension in S. At the end we also contribute to the study of the algebraic structure of E q (S) by testing it for the Riesz interpolation property. We show that E q (S) may fail to enjoy the Riesz interpolation property in both extreme situations when S is "very small" i.e. d( 2 /S) = 2 ℵ 0 and when S is 'very big' (i.e. d( 2 /S) = 2).

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“…We would like to summarize and refine our recent results in this area, bring new illustrating examples, and outline problems for future research. This contribution has grown out of the papers [3,4,5,6,9,10,11,12,14,15,16]. For basic treatment on this field, the reader can consult also monographs [7,8].…”

Section: Introductionmentioning

confidence: 99%

Probability structures in subspace lattice approach to foundations of quantum theory

Turilova

2015

Lith Math J

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Noncommutative measure and probability theory develops parallel to classical probability theory. In our paper, we summarize recent results on the structure of subspaces affiliated with von Neumann algebras and noncommutative measures defined on them. We show the interplay of ideas from classical and noncommutative measure theories based on general inner product spaces and bring new examples in this area.MSC: 46L10, 46L51, 46S10

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Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

Cited by 11 publications

References 10 publications

Quasi-splitting subspaces and Foulis-Randall subspaces

Quasi-splitting subspaces and Foulis-Randall subspaces

Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Probability structures in subspace lattice approach to foundations of quantum theory

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