Article
MSC:
03G12,
28A12,
46C05,
46N50,
81P10 | MR 2125153 | Zbl 1099.81010 | DOI: 10.1007/s10492-005-0007-1
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and $\sigma $-additive state
Hilbert space; inner product space; orthogonally closed subspace; complete and cocomplete subspaces; finitely and $\sigma $-additive state
Summary:
In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space $S$ is complete if and only if there exists a $\sigma $-additive state on $C(S)$, the orthomodular poset of complete-cocomplete subspaces of $S$. We then consider the problem of whether every state on $E(S)$, the class of splitting subspaces of $S$, can be extended to a Hilbertian state on $E(\bar{S})$; we show that for the dense hyperplane $S$ (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on $E(S)$ is a restriction of a state on $E(\bar{S})$.
References:
[1] J. Amemiya, H. Araki: A remark on Piron’s paper. Publ. Res. Inst. Math. Sci., Kyoto Univ., Ser. A 2 (1966), 423–427. DOI 10.2977/prims/1195195769 | MR 0213266
[2] G. Birkhoff, J. von Neumann: The logic of quantum mechanics. Ann. Math. 37 (1936), 823–843. DOI 10.2307/1968621 | MR 1503312
[3] A. Dvurečenskij: Gleason’s Theorem and Its Applications. Kluwer Acad. Publ., Ister Science Press, Dordrecht, Bratislava, 1993. MR 1256736
[4] A. Dvurečenskij, P. Pták: On states on orthogonally closed subspaces of an inner product space. Lett. Math. Phys. 62 (2002), 63–70. DOI 10.1023/A:1021653216049 | MR 1952116
[5] A. M. Gleason: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6 (1957), 885–893. MR 0096113 | Zbl 0078.28803
[6] J. Hamhalter, P. Pták: A completeness criterion for inner product spaces. Bull. London Math. Soc. 19 (1987), 259–263. DOI 10.1112/blms/19.3.259 | MR 0879514
[7] G. Kalmbach: Measures and Hilbert Lattices. World Sci. Publ. Co., Singapoore, 1986. MR 0867884 | Zbl 0656.06012
[8] P. Pták: ${\mathrm FAT}\leftrightarrow {\mathrm CAT}$ (in the state space of quantum logics). Proceedings of “Winter School of Measure Theory”, Liptovský Ján, Czechoslovakia, 1988, pp. 113–118. MR 1000201
[9] P. Pták, H. Weber: Lattice properties of subspace families in an inner product spaces. Proc. Am. Math. Soc. 129 (2001), 2111–2117. DOI 10.1090/S0002-9939-01-05855-5 | MR 1825924
Search
Partner of
