Abstract
We show that a finite orthomodular poset with a strong section Δ of states (probability measures) is distributive if and only if Δ has the unique Jordan-Hahn decomposition property(UJHDP). That this result does not extend to infinite orthomodular posets is shown by the projection lattices of von Neumann algebras without direct summand of typeI 2, for which the set of completely additive states is strong and has theUJHDP. There also exist nondistributive σ-classes for which the set of countably additive states has theUJHDP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Binder and M. Navara, “Quantum logics with lattice state spaces,”Proc. Am. Math. Soc. 100, 688–693 (1987).
A. Brøndsted,An Introduction to Convex Polytopes (Vol. 90 of Graduate Texts in Mathematics) (Springer, New York, 1983).
E. Christensen, “Measures on projections and physical states,”Commun. Math. Phys. 86, 329–338 (1982).
T. A. Cook, “The geometry of generalized quantum logics,”Int. J. Theor. Phys. 17, 941–955 (1978).
T. A. Cook and G. T. Rüttimann, “Symmetries on quantum logics,”Rep. Math. Phys. 21, 121–126 (1985).
J. C. Dacey, “Orthomodular spaces,” Dissertation, University of Massachusetts, 1968.
D. J. Foulis and C. H. Randall, “Operational statistics I. Basic concepts,”J. Math. Phys.,13, 1667–1675 (1972).
S. P. Gudder,Stochastic Methods in Quantum Mechanics (North-Holland, New York, 1979).
P. R. Halmos,Measure Theory (Van Nostrand-Reinhold, Princeton, 1950).
G. Kalmbach,Orthomodular Lattices (Academic Press, London, 1983).
M. P. Kläy, “Quantum logic properties of hypergraphs,”Found. Phys. 17, 1018–1036 (1987).
M. Navara, “State space properties of finite logics,”Czech. Math. J. 37, 188–196 (1987).
G. K. Pedersen,C*-Algebras and Their Automorphism Groups (Academic Press, London, 1979).
C. H. Randall and D. J. Foulis, “Operational statistics II. Manuals of operations and their logics,”J. Math. Phys. 14, 1472–1480 (1973).
G. T. Rüttimann, “Jordan-Hahn decomposition of signed weights on finite orthogonality spaces,”Comment. Math. Helv. 52, 129–144 (1977).
G. T. Rüttimann, “Facial sets of probability measures,”Prob. Math. Stat. 6, 99–127 (1985).
G. T. Rüttimann, “The approximate Jordan-Hahn decomposition,”Can. J. Math. 41, 1989.
S. Sakai,C*-Algebras and W*-Algebras (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 60) (Springer, Berlin, 1971).
C. Schindler, “Decompositions of measures on orthologics,” Dissertation, Universität Bern, 1986.
C. Schindler, “Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property,”Found. Phys. 19, 1299–1314 (1989).
F. J. Yeadon, “Measures on projections inW*-algebras of type II,”Bull. London Math. Soc. 15, 139–145 (1983).
F. J. Yeadon, “Finitely additive measures on projections in finiteW*-algebras,”Bull. London Math. Soc. 16, 145–150 (1984).
N. Zierler, “On general measure theory,” Dissertation, Harvard University, 1959.
Author information
Authors and Affiliations
Additional information
Research supported by Schweizerischer Nationalfonds/Fonds National Suisse under grant number 2.445-0.87.
Rights and permissions
About this article
Cite this article
Schindler, C. The unique Jordan-Hahn decomposition property. Found Phys 20, 561–573 (1990). https://doi.org/10.1007/BF01883239
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01883239