Abstract
We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Pták on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.
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Amemiya, I. and Araki, H.: A remark on Piron's paper, Publ. Res. Inst. Math. Sci. Kyoto Univ. Ser. A 2 (1966), 423–427.
Chetcuti, E.: Completeness criteria for inner product spaces, MSc thesis, Univ. of Malta, 2002.
Dvurečenskij, A.: Gleason's Theorem and Its Applications, Kluwer Acad. Publ., Dordrecht; Ister Science Press, Bratislava, 1992.
Dvurečenskij, A.: Frame functions and completeness of inner product spaces, Ann. Inst. H. Poincaré 62 (1995), 429–438.
Dvurečenskij, A. and Ptàk, P.: On states on orthogonally closed subspaces of an inner product space, Lett. Math. Phys. 62 (2002), 63–70.
Dvurečenskij, A., Neubrunn, T. and Pulmannovà, S.: Finitely additive states and completeness of inner product spaces, Found. Phys. 20 (1990), 1091–1102.
Gleason, A. M.: Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893.
Hamhalter, J. and Ptàk, P.: A completeness criterion for inner product spaces, Bull. London Math. Soc. 19 (1987), 259–263.
Ptàk, P.: FAT-CAT (in the state space of quantum logics), In: Proc. Winter School of Measure Theory, Liptovský, Jàn. 1988, pp. 113–118.
Ptàk, P. and Weber, H.: Two remarks on the subspace poset of an inner product space, In: Proc. Klagenfurt Conference, 19 May-1 June, 1997, Contributions to General Algebra 10, Verlag Johannes Heyn, Klagenfurt, 1998, pp. 263–267.
Maeda, F. and Maeda, S.: Theory of Symmetric Lattices, Springer, Berlin, 1970.
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Chetcuti, E., Dvurečenskij, A. A Finitely Additive State Criterion for the Completeness of Inner Product Spaces. Letters in Mathematical Physics 64, 221–227 (2003). https://doi.org/10.1023/A:1025757419340
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DOI: https://doi.org/10.1023/A:1025757419340