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Small Riesz spaces

  • G. Buskes (a1) and A. van Rooij (a2)
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Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

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[1] Abramovich, Y. A.. Multiplicative representation of disjointness preserving operators. Indag. Math. 45 (1983), 265297.
[2] Aliprantis, C. D. and Burkinshaw, O.. Positive Operators (Academic Press, 1985).
[3] Bernau, S. J.. Orthomorphisms of Archimedean vector lattices. Math. Proc. Cambridge Philos. Soc. 89 (1981), 119128.
[4] F, Beukers and Huijsmans, C. B.. Calculus in f-algebras. J. Austral. Math. Soc. Ser. A 37 (1984), 110116.
[5] Bishop, E. and Bridges, D.. Constructive Analysis (Springer-Verlag, 1985).
[6] Bigard, A., Keimel, K. and Wolfenstein, S.. Groupes et Anneaux réticulés. Lecture Notes in Math. vol. 608 (Springer-Verlag, 1977).
[7] Buskes, G. and Van Rooij, A.. Riesz spaces and the prime ideal theorem. (Preprint.)
[8] Duhoux, M. and Meyer, M.. A new proof of the lattice structure of orthomorphisms. J. London Math. Soc. (2) 25 (1982), 375378.
[9] Huijsmans, C. B. and de Pagter, B.. Ideal theory in f-algebras. Trans. Amer. Math. Soc. 269 (1982), 225245.
[10] Huijsmans, C. B. and de Pagter, B.. Subalgebras and Riesz subspaces of an f-algebra. Proc. London Math. Soc. (3) 48 (1984), 161174.
[11] Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3 (Cambridge University Press, 1982).
[12] Krivine, J. L.. Théorème de Factorisation dans les Espaces Réticulés. Séminaire Maurey-Schwartz, Exposés 2223 (1973).
[13] Kutateladze, S. S.. Support set for sublinear operators. Soviet Math. 17 (1976), 14281431.
[14] Lipecki, Z.. Extension of vector lattice homomorphisms. Proc. Amer. Math. Soc. 79 (1980), 247248.
[15] Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, vol. 2 (Springer-Verlag, 1979).
[16] Luxemburg, W. A.J., Some Aspects of the Theory of Riesz Spaces. Univ. Arkansas Lecture Notes Math. Sci. no. 4 (Wiley, 1979).
[17] Luxemburg, W. A. J. and Schep, A. R.. An extension theorem for Riesz homomorphisms. Indag. Math. 41 (1979), 145154.
[18] Luxemburg, W. A. J. and Zaanen, A. C.. Riesz Spaces, vol. 1 (North-Holland Publishing Company, 1971).
[19] McPolin, P. T. N. and Wickstead, A. W.. The order boundedness of band preserving operators on uniformly complete vector lattices. Math. Proc. Cambridge Philos. Soc. 97 (1985), 481487.
[20] de Pagter, B.. Calculus in f-algebras. (Preprint.)
[21] de Pagter, B.. f-algebras and orthomorphisms. Ph.D. thesis, Leiden University (1981).
[22] de Pagtee, B.. A note on disjointness preserving operators. Proc. Amer. Math. Soc. 90 (1984), 543549.
[23] Pták, V.. On a Theorem of Mazur and Orlicz. Studia Math. 15 (1956), 365366.
[24] Schep, A. R.. Positive diagonal and triangular operators. J. Operator Theory 3 (1980), 165178.
[25] Sikorski, R.. On a theorem of Mazur and Orlicz. Studia Math. 14 (1953), 180182.
[26] Simons, S.. Extended and sandwich versions of the Hahn-Banaeh theorem. J. Math. Anal. Appl. 21 (1968), 112122.
[27] Vulikh, B. Z.. Introduction to the Theory of Partially Ordered Spaces (Wolters-Noordhoff, 1967).
[28] Wickstead, A. W.. Extensions of orthomorphisms. J. Austral. Math. Soc. Ser. A 29 (1980), 8798.
[29] Zaanen, A. C.. Riesz Spaces, vol. 2 (North-Holland Publishing Company, 1983).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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