| Title:
|
On complete-cocomplete subspaces of an inner product space (English) |
| Author:
|
Buhagiar, David |
| Author:
|
Chetcuti, Emanuel |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
103-114 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $E(\bar{S})$. (English) |
| Keyword:
|
Hilbert space |
| Keyword:
|
inner product space |
| Keyword:
|
orthogonally closed subspace |
| Keyword:
|
complete and cocomplete subspaces |
| Keyword:
|
finitely and $\sigma $-additive state |
| MSC:
|
03G12 |
| MSC:
|
28A12 |
| MSC:
|
46C05 |
| MSC:
|
46N50 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 1099.81010 |
| idMR:
|
MR2125153 |
| DOI:
|
10.1007/s10492-005-0007-1 |
| . |
| Date available:
|
2009-09-22T18:21:07Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134595 |
| . |
| Reference:
|
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| Reference:
|
[2] G. Birkhoff, J. von Neumann: The logic of quantum mechanics.Ann. Math. 37 (1936), 823–843. MR 1503312, 10.2307/1968621 |
| Reference:
|
[3] A. Dvurečenskij: Gleason’s Theorem and Its Applications.Kluwer Acad. Publ., Ister Science Press, Dordrecht, Bratislava, 1993. MR 1256736 |
| Reference:
|
[4] A. Dvurečenskij, P. Pták: On states on orthogonally closed subspaces of an inner product space.Lett. Math. Phys. 62 (2002), 63–70. MR 1952116, 10.1023/A:1021653216049 |
| Reference:
|
[5] A. M. Gleason: Measures on the closed subspaces of a Hilbert space.J. Math. Mech. 6 (1957), 885–893. Zbl 0078.28803, MR 0096113 |
| Reference:
|
[6] J. Hamhalter, P. Pták: A completeness criterion for inner product spaces.Bull. London Math. Soc. 19 (1987), 259–263. MR 0879514, 10.1112/blms/19.3.259 |
| Reference:
|
[7] G. Kalmbach: Measures and Hilbert Lattices.World Sci. Publ. Co., Singapoore, 1986. Zbl 0656.06012, MR 0867884 |
| Reference:
|
[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 |
| Reference:
|
[9] P. Pták, H. Weber: Lattice properties of subspace families in an inner product spaces.Proc. Am. Math. Soc. 129 (2001), 2111–2117. MR 1825924, 10.1090/S0002-9939-01-05855-5 |
| . |
