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1998 | 158 | 2 | 109-124
Tytuł artykułu

Decomposition of group-valued measures on orthoalgebras

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We present a general decomposition theorem for a positive inner regular finitely additive measure on an orthoalgebra L with values in an ordered topological group G, not necessarily commutative. In the case where L is a Boolean algebra, we establish the uniqueness of such a decomposition. With mild extra hypotheses on G, we extend this Boolean decomposition, preserving the uniqueness, to the case where the measure is order bounded instead of being positive. This last result generalizes A. D. Aleksandrov's classical decomposition theorem.
Słowa kluczowe
Rocznik
Tom
158
Numer
2
Strony
109-124
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-01-20
poprawiono
1998-02-08
poprawiono
1998-04-27
Twórcy
  • Département de mathématiques et d'informatique, Université de Sherbrooke Sherbrooke, Québec J1K 2R1, Canada, anne.lacasse@dmi.usherb.ca
  • Dipartimento di Matematica e Applicazioni, Università Federico II, Complesso Universitario Monte S. Angelo, I-80126 Napoli, Italy, delucia@matna3.dma.unina.it
Bibliografia
  • [1] A. D. Alexandroff [A. D. Aleksandrov], Additive set-functions in abstract spaces, Part 1, Mat. Sb. 8 (50) (1940), 307-348.
  • [2] A. D. Alexandroff [A. D. Aleksandrov], Additive set-functions in abstract spaces, Part 2, ibid. 9 (51) (1941), 563-628.
  • [3] E. G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics, Addison-Wesley, Reading, Mass., 1981.
  • [4] A. Bigard, K. Keimel et S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math. 608, Springer, New York, 1977.
  • [5] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, 3rd ed., Providence, R.I., 1967.
  • [6] G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. of Math. 37 (1936), 823-843.
  • [7] P. De Lucia and P. Morales, Non-commutative version of the Alexandroff Decomposition Theorem in ordered topological groups, preprint no. 51, Univ. of Naples, 1993, 21 pp.
  • [8] A. Dvurečenskij, Gleason's Theorem and Its Applications, Kluwer, Dordrecht, 1993.
  • [9] A. Dvurečenskij and B. Riečan, Decomposition of measures on orthoalgebras and difference posets, Internat. J. Theoret. Phys. 33 (1994), 1387-1402.
  • [10] D. Feldman and A. Wilce, σ-Additivity in manuals and orthoalgebras, Order 10 (1993), 383-392.
  • [11] D. J. Foulis and M. K. Bennett, Tensor product of orthoalgebras, ibid., 271-282.
  • [12] D. J. Foulis, R. J. Greechie and G. T. Rüttimann, Filters and supports in orthalgebras, Internat. J. Theoret. Phys. 31 (1992), 789-807.
  • [13] F. Garcia-Mazario, Ordered topological group-valued measures on orthoalgebras, doctoral dissertation, UNED, 1995 (in Spanish).
  • [14] E. D. Habil, Brooks-Jewett and Nikodym convergence theorems for orthoalgebras that have the weak subsequential property, Internat. J. Theoret. Phys. 34 (1995), 465-491.
  • [15] G. Jameson, Ordered Linear Spaces, Lecture Notes in Math. 141, Springer, New York, 1970.
  • [16] G. Kalmbach, Orthomodular Lattices, Academic Press, London, 1983.
  • [17] J. Kelley, General Topology, Grad. Texts in Math. 27, Springer, New York, 1985.
  • [18] G. W. Mackey, The Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1963.
  • [19] S. Maeda, Probability measures on projections in von Neumann algebras, Rev. Math. Phys. 1 (1990), 235-290.
  • [20] P. Morales and F. Garcia-Mazario, The support of a measure in ordered topological groups, Atti Sem. Mat. Fis. Univ. Modena 45 (1997), 179-221.
  • [21] G. T. Rüttimann, Non-commutative measure theory, Habilitationsschrift, Universität Bern, 1980.
  • [22] G. T. Rüttimann, The approximate Jordan-Hahn decomposition, Canad. J. Math. 41 (1989), 1124-1146.
  • [23] K. Sundaresan and P. W. Day, Regularity of group valued Baire and Borel measures, Proc. Amer. Math. Soc. 36 (1972), 609-612.
  • [24] V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed., Springer, Berlin, 1985.
  • [25] K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46-66.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv158i2p109bwm
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