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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A topology on quantum logics


Authors: Sylvia Pulmannová and Zdena Riečanová
Journal: Proc. Amer. Math. Soc. 106 (1989), 891-897
MSC: Primary 81B10; Secondary 06C15, 54A99
MathSciNet review: 967488
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Abstract: A uniform topology $ {\tau _M}$ induced by a set $ M$ of finite measures on a quantum logic $ L$ is studied. If $ m$ is a valuation on $ L$, the topology $ {\tau _m}$ induced by $ \{ m\} $ is equivalent to the topology induced by the pseudometric $ \rho (a,b) = m(a\Delta b)$. If the set $ M$ of measures is large enough, the topology $ {\tau _M}$ reflects in some sense the structure of $ L$: if $ L$ is a continuous geometry and the measures are totally additive, $ {\tau _M}$ is weaker than the order topology $ {\tau _o}$ on $ L$. If $ L$ is atomic, $ {\tau _M}$ is stronger than $ {\tau _o}$. On a separable Hilbert space logic, $ {\tau _M}$ coincides with the discrete topology. Some cases are found in which $ {\tau _M} = {\tau _o}$.


References [Enhancements On Off] (What's this?)

  • [1] Ladislav Beran, Orthomodular lattices, Mathematics and its Applications (East European Series), D. Reidel Publishing Co., Dordrecht, 1985. Algebraic approach. MR 784029 (86m:06015b)
  • [2] G. Birkgof, Teoriya reshetok, “Nauka”, Moscow, 1984 (Russian). Translated from the English and with an appendix by V. N. Saliĭ; Translation edited by L. A. Skornyakov. MR 751233 (85e:06001)
  • [3] Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496 (85f:06012)
  • [4] Vladimír Palko, Topologies on quantum logics induced by measures, Math. Slovaca 39 (1989), no. 2, 175–189 (English, with Russian summary). MR 1018259 (90i:03071)
  • [5] Zdenka Riečanová, Topology in a quantum logic induced by a measure, Proceedings of the Conference: Topology and Measure, V (Binz, 1987), Wissensch. Beitr., Ernst-Moritz-Arndt Univ., Greifswald, 1988, pp. 126–130. MR 1029570 (91b:81015)
  • [6] T. A. Sarymsakov, S. A. Ajupov, Z. Chadžijev, and V. J. čilin, Uporiadočennije algebry, Taškent, "FAN" 1983.
  • [7] V. S. Varadarajan, Geometry of quantum theory, 2nd ed., Springer-Verlag, New York, 1985. MR 805158 (87a:81009)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0967488-4
PII: S 0002-9939(1989)0967488-4
Keywords: Orthomodular lattice, quantum logic, uniform topology generated by measures
Article copyright: © Copyright 1989 American Mathematical Society