Completeness of Inner Product Spaces and G. Cattaneo

• Autorzy: Buhagiar D. , Chetcuti E. , Dvurečenskij A.
• Języki publikacji: EN

Abstrakty

EN

In the Eighties G. Cattaneo contributed to Hilbert space quantummechanicsmodels using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the last 25 years into a rich theory which has a close connection to quantum information. G. Cattaneo also posed an open problem which was solved only this year.

Słowa kluczowe

EN

orthomodular lattices; algebraic completeness criteria

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2012
• Tom: Vol. 115, nr 1
• Strony: 15--24
• Opis fizyczny: Bibliogr.30 poz.

Twórcy

autor

Buhagiar D.

autor

Chetcuti E.

autor

Dvurečenskij A.

• Institute of Mathematics, Slovak Academy of Sciences, Štefanikova 49, SK-814 73 Bratislava, Slovakia,

Bibliografia

• [1] Amemiya, I., Araki, H.: A remark on Piron's paper, Publ. Res. Inst. Math. Sci., Kyoto Univ. Ser. A 2, 1966-67, 423-427.
• [2] Birkhoff, G., von Neumann, J.: The logic of quantum mechanics, Ann. Math., 37, 1936, 823-843.
• [3] Buhagiar, D., Chetcuti, E.: Quasi-splitting subspaces in a pre-Hilbert space, Math. Nachr., 280, 2007, 479-484.
• [4] Buhagiar, D., Chetcuti, E.: Only 'free' measures are admissable on F(S) when the inner product space is incomplete, Proc. Amer. Math. Soc., 136, 2008, 919-922.
• [5] Buhagiar, D., Chetcuti, E., Dvurečenskij, A.: Algebraic and measure-theoretic properties of classes of subspaces of an inner product space, in: Handbook of Quantum Logic and Quantum Structures, Quantum Structures, (K. Engesser, Dov M. Gabbay, D. Lehmann, Eds), Elsevier, 2007, 75-120.
• [6] Buhagiar, D., Chetcuti, E., Dvurečenskij, A.: Quasi-splitting subspaces and Foulis-Randall subspaces, J. Math. Phys., 52, (12) 2011, 123508-07.
• [7] Buhagiar, D., Chetcuti, E., Weber, H.: Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces, J. Math. Anal. Appl., 345, (2008), 725-730.
• [8] Cattaneo, G., Franco, G., Marino, G.: Ordering on families of subspaces of pre-Hilbert spaces and Dacey pre-Hilbert spaces, Boll. Unione Mat. Ital., (1-B) 1987, 167-183.
• [9] Cattaneo, G., Marino, G.: Spectral decomposition of pre-Hilbert spaces as regard to suitable classes of normal closed operators, Boll. Un. Mat. Ital. 6 (1-B) 1982, 451-466.
• [10] Cattaneo, G., Marino, G.: Alcuni interessanti posets di sottospazzi di uno spazio pre-Hilbert, Rendi. Semin. Math. Fis. Milano, 52 1983, 69-74.
• [11] Cattaneo, G., Marino, G.: Completeness of inner product spaces with respect to splitting subspaces, Letters Math. Phys., 11 1986, 15-20.
• [12] Chetcuti, E., Dvurečenskij, A.: A finitely additive state criterion for the completeness of inner product spaces, Letters Math. Phys., 64, 2003, 221-227.
• [13] Chetcuti, E., Dvurečenskij, A.: The existence of finitely additive states on orthogonally closed subspaces of incomplete inner product spaces, Letters Math. Phys., 67, 2004, 75-80.
• [14] Chetcuti, E., Dvurečenskij, A.: The state-space of the lattice of orthogonally closed subspaces, Glasgow Math. J., 47, 2005, 213-220.
• [15] Dalla Chiara,M., Giuntini, R., Greechie, R.: "Reasoning in Quantum Theory. Sharpened Unsharp Quantum Logics", Kluwer Academic Publ., Dordrecht, 2004.
• [16] Dvurečenskij, A.: Completeness of inner product spaces and quantum logic of splitting subspaces, Letters Math. Phys. 15 1988, 231-235.
• [17] Dvurečenskij, A.: "Gleason's Theorem and Its Applications", Kluwer Academic Publisher, Dordrecht/Boston/London, 1993, 325+xv pp.
• [18] Dvurečenskij, A.: Frame functions and completeness of inner product spaces, Ann. Inst. Henri Poincaré Phys. Théor., 62, 1995, 429-438.
• [19] Dvurečenskij, A.: A new algebraic criterion for completeness of inner product spaces, Letters Math. Phys., 58, 2001, 205-208.
• [20] Dvurečenskij, A., Pták, P.: On states on orthogonally closed subspaces of an inner product space, Letters Math. Phys., 62 2002, 63-70.
• [21] Dvurečenskij A., Pulmannová, S.: "New Trends in Quantum Structures", Kluwer Acad. Publ., Dordrecht, 2000.
• [22] Gleason, A.M.: Measures on the closed subspaces of a Hilbert space, J. Math. Mech., 6 1957, 885-893.
• [23] Gross, H., Keller, H.A.: On the definition of Hilbert space, Manuscr. Math., 23, 1977, 67-90.
• [24] Gudder, S.P.: Correction to "Inner product spaces", Amer. Math. Monthly, 82, 1975, 251-252.
• [25] Hamhalter, J.: Quantum Measure Theory, Kluwer Acad. Publ., Dordrecht, 2003.
• [26] Hamhalter, J., Pták, P.: A completeness criterion for inner product spaces, Bull. London Math. Soc., 19 1987, 259-263.
• [27] Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin, 1933.
• [28] Piron, C.: Axiomatique quantique, Helv. Phys. Acta, 37, 1964, 439-468.
• [29] Pták, P.: FAT - CAT (in the state space of quantum logics), in: Proceedings of "Winter School of Measure Theory", Liptovský Ján 1988, (Czechoslovakia), 113-118.
• [30] Pták, P.,Weber, H.: Lattice properties of subspace families of inner product spaces, Proc. Amer. Math. Soc., 129, 2001, 2111-2117.