An orthomodular lattice admitting no group-valued measure
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- by Mirko Navara PDF
- Proc. Amer. Math. Soc. 122 (1994), 7-12 Request permission
Abstract:
We construct a finite orthomodular lattice L such that, for each commutative group G, there is no nontrivial G-valued measure on L. This result extends a result of R. J. Greechie (Orthogonal lattices admitting no states, J. Combin. Theory Ser. A 10 (1971), 119-132), and also sheds light on recent investigations in the noncommutative measure theory.References
- James K. Brooks and Robert S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298. MR 269802, DOI 10.1073/pnas.67.3.1294
- Anna Bruna D’Andrea and Paolo De Lucia, The Brooks-Jewett theorem on an orthomodular lattice, J. Math. Anal. Appl. 154 (1991), no. 2, 507–522. MR 1088647, DOI 10.1016/0022-247X(91)90054-4 P. de Lucia and P. Morales, A non-commutative version of a theorem of Marczewski for submeasures, Studia Math. (to appear). P. de Lucia and T. Traynor, Non-commutative group-valued measures on an orthomodular poset (to appear).
- R. J. Greechie, Orthomodular lattices admitting no states, J. Combinatorial Theory Ser. A 10 (1971), 119–132. MR 274355, DOI 10.1016/0097-3165(71)90015-x R. J. Greechie and F. R. Miller, On structures related to states on an empirical logic I. Weights on finite spaces, Technical Report 16, Dept. of Math., Kansas State Univ., Manhattan, KA, 1970.
- Stanley P. Gudder, Stochastic methods in quantum mechanics, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York-Oxford, 1979. MR 543489
- Jan Hamhalter and Pavel Pták, Hilbert-space-valued states on quantum logics, Appl. Math. 37 (1992), no. 1, 51–61. MR 1152157
- Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496 A. G. Kurosh., Group theory, Nauka, Moskow, 1967. (Russian)
- Mirko Navara, Pavel Pták, and Vladimír Rogalewicz, Enlargements of quantum logics, Pacific J. Math. 135 (1988), no. 2, 361–369. MR 968618
- Mirko Navara and Vladimír Rogalewicz, Construction of orthomodular lattices with given state spaces, Demonstratio Math. 21 (1988), no. 2, 481–493. MR 981700
- Mirko Navara and Vladimír Rogalewicz, State isomorphism of orthomodular posets and hypergraphs, Proceedings of the First Winter School on Measure Theory (Liptovský Ján, 1988) Slovak Acad. Sci., Bratislava, 1988, pp. 93–98. MR 1000196
- M. Navara and G. T. Rüttimann, A characterization of $\sigma$-state spaces of orthomodular lattices, Exposition. Math. 9 (1991), no. 3, 275–284. MR 1121158
- Pavel Pták, Exotic logics, Colloq. Math. 54 (1987), no. 1, 1–7. MR 928651, DOI 10.4064/cm-54-1-1-7 P. Pták and S. Pulmannová, Orthomodular structures as quantum logics, Kluwer, Dordrecht, Boston, and London, 1991.
- Frederic W. Shultz, A characterization of state spaces of orthomodular lattices, J. Combinatorial Theory Ser. A 17 (1974), 317–328. MR 364042, DOI 10.1016/0097-3165(74)90096-x
- V. S. Varadarajan, Geometry of quantum theory. Vol. I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0471674
- Hans Weber, Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodým’s boundedness theorem, Rocky Mountain J. Math. 16 (1986), no. 2, 253–275. MR 843053, DOI 10.1216/RMJ-1986-16-2-253
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 7-12
- MSC: Primary 06C15; Secondary 03G12, 28B10
- DOI: https://doi.org/10.1090/S0002-9939-1994-1191871-X
- MathSciNet review: 1191871