| Title:
|
Group-valued measures on coarse-grained quantum logics (English) |
| Author:
|
de Simone, Anna |
| Author:
|
Pták, Pavel |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
737-746 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later () this result was re-proved (and further improved on) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof technique is entirely different from that of the preceding papers. In addition, we provide a new combinatorial argument for describing all atoms of cyclic coarse-grained quantum logics. (English) |
| Keyword:
|
coarse-grained quantum logic |
| Keyword:
|
group-valued measure |
| Keyword:
|
measure extension |
| MSC:
|
03G12 |
| MSC:
|
06C15 |
| MSC:
|
28A55 |
| MSC:
|
28A99 |
| MSC:
|
28B10 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 1174.03350 |
| idMR:
|
MR2337627 |
| . |
| Date available:
|
2009-09-24T11:49:02Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128202 |
| . |
| Reference:
|
[1] M. R. Darnel: Theory of Lattice-Ordered Groups.Dekker, New York, 1995. Zbl 0810.06016, MR 1304052 |
| Reference:
|
[2] A. De Simone, M. Navara and P. Pták: Extensions of states on concrete finite logics.(to appear). |
| Reference:
|
[3] S. Gudder and J. P. Marchand: A coarse-grained measure theory.Bull. Polish Acad. Sci. Math. 28 (1980), 557–564. MR 0628642 |
| Reference:
|
[4] S. Gudder: Stochastic Methods in Quantum Mechanics.North Holland, 1979. Zbl 0439.46047, MR 0543489 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[7] P. de Lucia and P. Pták: Quantum logics with classically determined states.Colloq. Math. 80 (1999), 147–154. MR 1684578, 10.4064/cm-80-1-147-154 |
| Reference:
|
[8] M. Navara and P. Pták: Almost Boolean orthomodular posets.J. Pure Appl. Algebra 60 (1989), 105–111. MR 1014608, 10.1016/0022-4049(89)90108-4 |
| Reference:
|
[9] P. Ovtchinikoff: Measures on the Gudder-Marchand logics.Constructive Theory of Functions and Functional Analysis 8 (1992), 95–98. (Russian) MR 1231108 |
| Reference:
|
[10] P. Pták: Some nearly Boolean orthomodular posets.Proc. Amer. Math. Soc. 126 (1998), 2039–2046. MR 1452822, 10.1090/S0002-9939-98-04403-7 |
| Reference:
|
[11] P. Pták: Concrete quantum logics.Internat. J. Theoret. Phys. 39 (2000), 827–837. MR 1792201, 10.1023/A:1003626929648 |
| Reference:
|
[12] P. Pták and S. Pulmannová: Orthomodular Structures as Quantum Logics.Kluwer, Dordrecht/Boston/London, 1991. MR 1176314 |
| . |
