Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Noncommutative decomposition theorems in Riesz spaces


Authors: Paolo De Lucia and Pedro Morales
Journal: Proc. Amer. Math. Soc. 120 (1994), 193-202
MSC: Primary 28B05; Secondary 03C15, 06A06, 46L50
MathSciNet review: 1203982
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Abstract: We show that an additive function defined on an orthomodular poset and taking its values in the positive cone of a normed Riesz space admits a Lebesgue Decomposition and a Yosida-Hewitt Decomposition.


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  • [1] J. F. Aarnes, Quasi-states on $ {C^{\ast}}$-algebras, Trans. Amer. Math. Soc. 149 (1970), 601-625. MR 0282602 (43:8311)
  • [2] L. Beran, Orthomodular lattices. Algebraic approach, Academia, Praha, and Reidel, Dordrecht, 1984. MR 785005 (86m:06015a)
  • [3] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges, Academic Press, London, 1983. MR 751777 (86f:28006)
  • [4] N. Bourbaki, Topologie générale, Chapitres 1-4, Hermann, Paris, 1971. MR 0358652 (50:11111)
  • [5] W. J. Claas and A. C. Zaanen, Orlicz lattices, Compositio Math., Tomus specialis in honorem Ladislai Orlicz (1978), 77-93. MR 504154 (80b:46010)
  • [6] R. Darst, A decomposition of finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31-37. MR 0137808 (25:1257)
  • [7] J. Dixmier, Les algèbres d' opérateurs dans l'espace Hilbertien (Algèbres de von Neumann), Gauthier-Villars, Paris, 1969.
  • [8] N. Dunford and J. Schwartz, Linear operators. I, Interscience, New York, 1958.
  • [9] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge Univ. Press, Cambridge, 1974. MR 0454575 (56:12824)
  • [10] P. Halmos, Introduction to Hilbert spaces and the theory of spectral multiplicity, Chelsea, New York, 1951. MR 1653399 (99g:47001)
  • [11] G. Kalmbach, Orthomodular lattices, Academic Press, London, 1983. MR 716496 (85f:06012)
  • [12] P. Ptak and S. Pulmannova, Orthomodular structures as quantum logics, Kluwer Academic, Dordrecht, 1991. MR 1176314 (94d:81018b)
  • [13] H. L. Royden, Real analysis, 2nd ed., MacMillan, New York, 1971. MR 0151555 (27:1540)
  • [14] G. T. Rüttimann, Decomposition of cones of measures, Atti Sem. Mat. Fis. Univ. Modena 28 (1990), 109-121. MR 1076452 (91m:28004)
  • [15] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin, 1974. MR 0423039 (54:11023)
  • [16] K. D. Schmidt, Decomposition of vector measures in Riesz spaces and Banach lattices, Proc. Edinburgh Math. Soc. (2) 29 (1986), 23-39. MR 829177 (87f:46075)
  • [17] K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46-66. MR 0045194 (13:543b)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1203982-0
Keywords: Orthomodular poset, normed Riesz space, Hilbert space, additive function, completely additive function
Article copyright: © Copyright 1994 American Mathematical Society