Weakly Purely Finitely Additive Measures

Rüttimann, Gottfried T.

Canadian journal of mathematics, Tome 46 (1994) no. 4, pp. 872-885

Résumé

Let L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additivemeasure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measurev and a weakly purely finitely additive measure ξ. We give sufficient conditions for thisYosida-Hewitt-type decomposition to be unique.A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

Détail

DOI : 10.4153/CJM-1994-049-1

Mots-clés : 28A60, 06C15, 81P10, orthomodular lattice, completely additive measure, weakly purely finitely additivemeasure, filtering measure, decomposition of measures

Bibliographie

[1] 1. Alfsen, E. M., Compact Convex Sets and Boundary Integrals, Springer Verlag, Berlin, 1971. Google Scholar

[2] 2. Battaglia, M., Annihilators in JB-Algebras, Math. Proc. Cambridge Philps. Soc. 108(1990), 317–323. Google Scholar

[3] 3. Birkhoff, G., Lattice Theory, Third edition, Amer. Math. Soc, Providence R.I., 1967. Google Scholar

[4] 4. Bunce, L. J. and Wright, J. D. M., Continuity and Linear Extensions of Quantum Measures on Jordan Operator Algebras, Math. Scand. 64(1989), 300ߝ306. Google Scholar

[5] 5. Edwards, C. M. and Riittimann, G. T., On the Facial Structure of the Unit Balls in a GL-Space and its Dual, Math. Proc. Cambridge Philos. Soc. 98(1985), 305–322. Google Scholar

[6] 6. Greechie, R. J., Orthomodular Lattices Admitting No States, J. Combin. Theory Ser. A 10(1971), 119–132. Google Scholar

[7] 7. Hanche-Olsen, H. and Størmer, E., Jordan Operator Algebras, Pitman, Boston, 1984. Google Scholar

[8] 8. Kalmbach, G., Orthomodular Lattices, Academic Press, London, 1983. Google Scholar

[9] 9. Navara, M. and Rüttimann, G. T., A Characterization of σ-State Spaces of Orthomodular Lattices, Exposition. Math. 9(1991), 275–284. Google Scholar

[10] 10. Rüttimann, G. T., Lecture Notes on Base Normed and Order Unit Normed Spaces, University of Denver, 1984. Google Scholar

[11] 11. Rüttimann, G. T., Facial Sets of Probability Measures, Probab. Math. Statist. (Wrocłav) 6(1985), 187–215. Google Scholar

[12] 12. Rüttimann, G. T. and Schindler, Chr., The Lebesgue Decomposition of Measures on Orthomodular Posets, Quart. J. Math. Oxford Ser. (2) 37(1986), 321–345. Google Scholar

[13] 13. Rüttimann, G. T. and Schindler, Chr., On a-convex Sets of Probability Measures, Bull. Polish Acad. Sci. Math. 35(1987), 583–595. Google Scholar

[14] 14. Rüttimann, G. T., The Approximate Jordan-Hahn Decomposition, Canad. J. Math. 41(1989), 1124–1146. Google Scholar

[15] 15. Rüttimann, G. T., Decomposition of Cones of Measures, Atti Sem. Mat. Fis. Univ. Modena 38(1990), 267–269. Google Scholar

[16] 16. Takesaki, M., Theory of Operator Algebras I, Springer Verlag, Berlin, 1979. Google Scholar

[17] 17. Yosida, K. and Hewitt, E., Finitely Additive Measures, Trans. Amer. Math. Soc. 72(1952), 46–66. Google Scholar