Abstract
We show that an inner product space S is complete whenever the system E(S) of all splitting subspaces of S, i.e., of all subspaces M of S such that M + M ⊥ = S holds, satisfies the σ-Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a σ-complete lattice, and that of the author who required E(S) to be a σ-orthocomplete OMP.
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Amemiya, I. and Araki, H.: A remark on Piron's paper, Publ. Res. Inst.Math. Sci, Ser. A 12 (1966-67), 423-427.
Cattaneo, G. and Marino, G.: Completeness of inner product spaces with respect to splitting subspaces, Lett. Math. Phys. 11 (1986), 15-20.
Dvurečenskij, A.: Completeness of inner product spaces and quantum logic of splitting subspaces, Lett. Math. Phys. 15 (1988), 231-235.
Dvurečenskij, A.: Gleason's Theorem and Its Applications, Kluwer Acad. Publ., Dordrecht, Ister Science Press, Bratislava, 1992.
Gross, H. and Keller, H.: On the definition of Hilbert space, Manuscripta Math. 23 (1977), 67-90.
Pták, P. and Weber, H.: Lattice properties of closed subspaces of inner product spaces, Proc. Amer. Math. Soc. 129 (2001), 2111-2117.
Varadarajan, V. S.: Geometry of Quantum Theory, Vol 1, Van Nostrand, Princeton, 1968.
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Dvurečenskij, A. A New Algebraic Criterion for Completeness of Inner Product Spaces. Letters in Mathematical Physics 58, 205–208 (2001). https://doi.org/10.1023/A:1014500410095
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DOI: https://doi.org/10.1023/A:1014500410095