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Copies of \(c_{0}\) and \(\ell_{\infty}\) in topological Riesz spaces. (English) Zbl 0903.46010
Summary: The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of \(c_{0}\) or \(\ell_{\infty}\).

MSC:
46A40 Ordered topological linear spaces, vector lattices
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B42 Banach lattices
28B05 Vector-valued set functions, measures and integrals
40A99 Convergence and divergence of infinite limiting processes
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[1] Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics, Vol. 76. · Zbl 0402.46005
[2] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0608.47039
[3] G. Buskes and I. Labuda, On Levi-like properties and some of their applications in Riesz space theory, Canad. Math. Bull. 31 (1988), no. 4, 477 – 486. · Zbl 0631.46006
[4] A. Costé, Convergence des séries dans les espaces \?-normés de fonctions mesurables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 131 – 134 (French, with English and Russian summaries). · Zbl 0233.46037
[5] S. Díaz, On Schwartz’s C-spaces and Orlicz’s O-spaces, Colloq. Math. 64 (1993), 245-251. · Zbl 0821.46007
[6] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 725 – 731 (English, with Russian summary). · Zbl 0243.28011
[7] L. Drewnowski, On subseries convergence in some function spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 797 – 803 (English, with Russian summary). · Zbl 0297.28014
[8] Lech Drewnowski, Un théorème sur les opérateurs de \?_{\infty }(\Gamma ), C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 22, Aii, A967 – A969 (French, with English summary). · Zbl 0323.46014
[9] L. Drewnowski and I. Labuda, The Orlicz-Pettis theorem for topological Riesz spaces, Proc. Amer. Math. Soc. (to appear). · Zbl 0885.40002
[10] -, Spaces of Bochner measurable functions, Preprint (1996).
[11] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. · Zbl 0273.46035
[12] N. J. Kalton, Exhaustive operators and vector measures, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 291 – 300. · Zbl 0302.47020
[13] L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. · Zbl 0484.46003
[14] Leonard Y. H. Yap, On a convolution theorem, Acta Sci. Math. (Szeged) 38 (1976), no. 1-2, 203 – 204. · Zbl 0295.43004
[15] Iwo Labuda, Spaces of measurable functions, Comment. Math. Special Issue 2 (1979), 217 – 249. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. · Zbl 0477.46026
[16] Iwo Labuda, Submeasures and locally solid topologies on Riesz spaces, Math. Z. 195 (1987), no. 2, 179 – 196. · Zbl 0601.46006
[17] E. Langford and C. D. Aliprantis, Regularity properties of quotient Riesz seminorms, Nederl. Akad. Wetensch. Proc. Ser. A 78=Indag. Math. 37 (1975), 199 – 212. · Zbl 0306.46013
[18] W. Matuszewska and W. Orlicz, A note on modular spaces. IX, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 801 – 808 (English, with Loose Russian summary). · Zbl 0164.43002
[19] W. Orlicz, Über die Divergenz von allgemeinen Orthogonalreihen, Studia Math. 4 (1933), 27-32. · JFM 59.1012.01
[20] -, On a class of asymptotically divergent sequences of functions, Studia Math. 12 (1951), 286-307. · Zbl 0044.05701
[21] Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13 – 36. · Zbl 0227.46027
[22] Laurent Schwartz, Un théorème de convergence dans les \?^{\?}, 0\le \?<+\infty , C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A704 – A706 (French). · Zbl 0172.40003
[23] A. I. Veksler and V. A. Geĭler, Order completeness and disjoint completeness of linear partially ordered spaces, Sibirsk. Mat. Ž. 13 (1972), 43 – 51 (Russian). · Zbl 0333.06009
[24] Witold Wnuk, Locally solid Riesz spaces not containing \?\(_{0}\), Bull. Polish Acad. Sci. Math. 36 (1988), no. 1-2, 51 – 56 (English, with Russian summary). · Zbl 0676.46002
[25] Wojbor A. Woyczyński, Sur la convergence des séries dans les espaces de type (\?), C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1254 – A1257 (French). · Zbl 0175.42402
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