Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T18:23:45.230Z Has data issue: false hasContentIssue false
  • English
  • Français

The Approximate Jordan-Hahn Decomposition

Published online by Cambridge University Press:  20 November 2018

Gottfried T. Rüttimann
Gottfried T. Rüttimann*
Affiliation:
University of Berne, Berne, Switzerland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Non-commutative measure theory embraces measure theory on cr-fields of subsets of a set, on projection lattices of von Neumann algebras or JBW-algebras and on hypergraphs alike [20], [27], [33], [37], [39], [40], [41]. Due to the unifying structure of an orthoalgebra concepts can easily be transferred from one branch to the other. Additional conceptual inpetus is obtained from the logico-probabilistic foundations of quantum mechanics (see [6], [19], [21]).

In the late seventies the author studied the Jordan-Hahn decomposition of measures on orthomodular posets and certain graphs. These investigations revealed an interesting geometrical aspect of this decomposition in that the Jordan-Hahn property of the convex set of probability charges on a finite orthomodular poset can be characterized in terms of the extreme points of the unit ball of the Banach space dual of the base normed space of Jordan charges.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 6 , 01 December 1989 , pp. 1124 - 1146
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Alfsen, E. M., Compact convex sets and boundary integrals, (Springer Verlag, Berlin, 1971).Google Scholar
2. Alfsen, E. M. and Shultz, F. W., Non-commutative spectral theory for affine function spaces on convex sets, Memoirs of the American Mathematical Society 172 (1976).Google Scholar
3. Alfsen, E. M., Schultz, F. W. and E. St0rmer, A Gelfand-Neumark theorem for Jordan algebras, Advances in Mathematics 28 (1978).Google Scholar
4. Asimow, L.and Ellis, A. J., Convexity theory and its applications in functional analysis (Academic Press, London, 1980).Google Scholar
5. Birkhoff, G., Lattice theory, AMS Colloquium Publications 25, 3rd edition (American Mathematical Society, Providence, 1967).Google Scholar
6. Beltrametti, E. G. and Cassinelli, G., The logic of quantum mechanics, Encyclopedia of Mathematics and its Applications 75 (Addison-Wesley, Reading, 1981).Google Scholar
7. Berge, C., Graphs and hypergraphs (North Holland, New York, 1973).Google Scholar
8. Bunce, L. J. and Wright, J. D. M., Quantum measures and states on Jordan algebras, Communications in Mathematical Physics 98 (1985).Google Scholar
9. Cook, T. A., The geometry of generalized quantum logic, International Journal of Theoretical Physics 17 (1978).Google Scholar
10. Cook, T. A. and G. T. Rüttimann, Symmetries on quantum logics, Reports on Mathematical Physics 21 (1985).Google Scholar
11. Dilworth, R. P., On complemented lattices, Tôhoku Mathematical Journal 47 (1940).Google Scholar
12. Diestel, J., Geometry of Banach spaces - selected topics, Lecture Notes in Mathematics 485 (Springer-Verlag, Berlin, 1975).Google Scholar
13. Edwards, C. M. and Riittimann, G. T., On the facial structure of the unit balls in a GL-space and its dual, Mathematical Proceedings of the Cambridge Philosophical Society 98 (1985).Google Scholar
14. Edwards, C. M. and Riittimann, G. T., On the facial structure of the unit ball of a GM-space, Mathematische Zeitschrift 193 (1986).Google Scholar
15. Ellis, A. J., The duality of partially ordered normed linear spaces, Journal of the London Mathematical Society 39 (1964).Google Scholar
16. Foulis, D. J., A note on orthomodular lattices, Portugaliae Mathematica 21 (1962).Google Scholar
17. Foulis, D. J. and Randall, C. H., New definitions and results, Ditto Notes, University of Massachusetts, Autumn (1979).Google Scholar
18. Greechie, R. J., Orthomodular lattices admitting no states, Journal of Combinatorial Theory 4 (1971).Google Scholar
19. Gudder, S. P., Stochastic methods in quantum mechanics (North Holland, New York, 1979).Google Scholar
20. Gudder, S. P., Kläy, M. P. and Rüttimann, G. T., States on hyper graphs, Demonstratio Mathematica/ 9 (1986).Google Scholar
21. Gudder, S. P., Quantum probability (Academic Press, Boston, 1988).Google Scholar
22. Halmos, P. R., Boolean algebras (Van Nostrand Company, Princeton, 1963).Google Scholar
23. Hanche-Olson, H. and E. Størmer, , Jordan operator algebras (Pitman, Boston, 1984).Google Scholar
24. Holmes, R. B., Geometrical functional analysis and its applications (Springer-Verlag, Berlin, 1975).Google Scholar
25. James, R., Weak compactness and reflexivity, Israel Journal of Mathematics 2 (1964).Google Scholar
26. Janowitz, M. F., Quantifiers on quasi-orthomodular lattices, Doctoral Thesis, Wayne State University (1963).Google Scholar
27. Kalmbach, G., Measures and Hilbert lattices (World Scientific Publishing Company, Singapore, 1986).Google Scholar
28. Kläy, M. P., Stochastic models and empirical systems, empirical logics and quantum logics, and states on hypergraphs, Doctoral Thesis, University of Berne (1985).Google Scholar
29. Köthe, G., Topological vector spaces 1 (Springer-Verlag, 1969).Google Scholar
30. Lock, P. F. and Hardegree, G. M., Connections among quantum logics. Part 1 : Quantum prepositional logics, International Journal of Theoretical Physics 24 (1984).Google Scholar
31. Lock, P. F. and Hardegree, G. M., Connections among quantum logics. Part 2: Quantum event logics, International Journal of Theoretical Physics 24 (1984).Google Scholar
32. Randall, C. H. and Foulis, D. J., Operational statistics and tensor products, In nterpretations and foundations of quantum theory (Bibliographisches Institut, Zurich, 1981).Google Scholar
33. Stochastic entities, In Recent developments in quantum logics (Bibliographisches Institut, Zurich, 1985).Google Scholar
34. Randall, C. H. and Foulis, D. J., B, K. P. S.. Rao and Rao, M. B., Theory of charges (Academic Press, New York, 1983).Google Scholar
35. Rüttimann, G. T., Stable faces of a polytope, Bulletin of the American Mathematical Society 82 (1976).Google Scholar
36. Rüttimann, G. T., Jordan-Hahn decomposition of signed weights on finite orthogonality spaces, Commentarii Mathematici Helvetici 52 (1977).Google Scholar
37. Rüttimann, G. T., Non-commutative measure theory (Habilitationsschrift, Universitàt Bern, 1980).Google Scholar
38. Rüttimann, G. T., Lecture notes on base normed and order unit normed spaces (University of Denver, 1981/84).Google Scholar
39. Rüttimann, G. T., Facial sets of probability measures, Probability and Mathematical Statistics 6 (1985).Google Scholar
40. Rüttimann, G. T. and Chr. Schindler, The Lebesgue decomposition of measures on orthomodular posets, Quarterly Journal of Mathematics Oxford 37 (1986).Google Scholar
41. Rüttimann, G. T. and Chr. Schindler, On a-convex sets of probability measures, Bulletin of the Polish Academy of Science, Mathematics, 35 (1987).Google Scholar
42. Rüttimann, G. T., Orthoalgebras. Preprint, University of Berne (1989).Google Scholar
43. Schindler, Chr., Decomposition of measures on orthologics, Doctoral Thesis, University of Berne (1986).Google Scholar
44. Schultz, F. W., On normed Jordan algebras which are Banach space duals, Journal of Functional Analysis 31 (1979).Google Scholar
45. Zierler, N., On general measure theory, Doctoral Thesis, Harvard University (1959).Google Scholar