×
Filter, Wolfgang

Representations of Riesz spaces as spaces of measures. I. (English) Zbl 0803.46008

Czech. Math. J. 42, No. 3, 415-432 (1992).
The author gives a concrete representation for spaces which reflect many properties of duality. Especially he shows that a Riesz space \(E\) can be represented as an order dense Riesz subspace of a band \(M\) of measures iff it is represented by the set \(E^ \pi\) of its order continuous linear forms. \(E\) is an ideal of \(M\) iff \(E\) is Dedekind complete, \(E= M\) iff \(E\) is hypercomplete. The results of the author are also generalizations of some previous results by D. H. Fremlin [Proc. Camb. Philos. Soc. 63, 951-956 (1967; Zbl 0179.170)] and C. Constantinescu [‘Duality in measure theory’, Lecture Notes Math. 796 (1980; Zbl 0429.28009)].
Reviewer: B.Riečan (Bratislava)

Cited in 1 Review
Cited in 2 Documents

MSC:

46A40 Ordered topological linear spaces, vector lattices
46E27 Spaces of measures

Keywords:

order dense Riesz subspace of a band of measures; representation; duality; Riesz space; ideal; Dedekind complete

Citations:

Zbl 0803.46009; Zbl 0179.170; Zbl 0429.28009
PDFBibTeX XMLCite
\textit{W. Filter}, Czech. Math. J. 42, No. 3, 415--432 (1992; Zbl 0803.46008)
Full Text: DOI EuDML

References:

[1] C. D. Aliprantis and O. Burkinshaw: Locally solid Riesz spaces. Academic Press, New York-San Francisco-London, 1978. · Zbl 0402.46005
[2] S. J. Bernau: Unique representation of Archimedean lattice groups and normal Archimedean lattice rings. Proc. London Math. Soc. (3) 15 (1965), 599-631. · Zbl 0134.10802 · doi:10.1112/plms/s3-15.1.599
[3] H. F. Bohnenblust and S. Kakutani: Concrete representation of \((M)\)-spaces. Ann. of Math. 42 (1941), 1025-1028. · Zbl 0061.24209 · doi:10.2307/1968779
[4] L. Brown and H. Nakano: A representation theorem for Archimedean linear lattices. Proc. Amer. Math. Soc. 17 (1966), 835-837. · Zbl 0149.33305 · doi:10.2307/2036263
[5] G. Buskes and A. van Rooij: Small Riesz spaces. Math. Proc. Cambridge Philos. Soc. 105 (1989), 523-536. · Zbl 0683.46013 · doi:10.1017/S0305004100077902
[6] C. Constantinescu: Duality in measure theory. LN in mathematics 796, Springer-Verlag, Berlin-Heidelberg-New York, 1980. · Zbl 0429.28009
[7] E. B. Davies: The Choquet theory and representation of ordered Banach spaces. Illinois J. Math. 13 (1969), 176-187. · Zbl 0165.46801
[8] W. A. Feldman and J. F. Porter: Banach lattices with locally compact representation spaces. Math. Z. 174 (1980), 233-239. · Zbl 0425.46016 · doi:10.1007/BF01161412
[9] W. A. Feldman and J. F. Porter: Uniqueness of representation spaces. Math. Z. 179 (1982), 213-217. · Zbl 0458.46013 · doi:10.1007/BF01214313
[10] W. Filter: A note on Archimedean Riesz spaces and their extended order duals. Libertas Math. 6 (1986), 101-106. · Zbl 0612.46007
[11] W. Filter: Atomical and atomfree elements of a Riesz space. Arch. Math. (Basel) 52 (1989), 580-587. · Zbl 0659.46004 · doi:10.1007/BF01237571
[12] W. Filter: Hypercomplete Riesz spaces. Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 227-240. · Zbl 0716.46008
[13] W. Filter: Hypercompletions of Riesz spaces. Proc. Amer. Math. Soc. 109 (1990), 775-780. · Zbl 0716.46009 · doi:10.2307/2048219
[14] W. Filter: Hellinger-type integrals in Riesz spaces. Houston J. Math. 17 (1991), 179-200. · Zbl 0778.46009
[15] W. Filter: Measurability and decomposition properties in the dual of a Riesz space. Rend. Circ. Mat. Palermo, Suppl. Volume · Zbl 0780.46006
[16] I. Fleischer: Functional representation of vector lattices. Proc. Amer. Math. Soc. 108 (1990), 471-478. · Zbl 0743.46005 · doi:10.2307/2048297
[17] D. H. Fremlin: Abstract Köthe spaces. II. Proc. Cambridge Philos. Soc. 63 (1967), 951-956. · Zbl 0179.17005 · doi:10.1017/S0305004100041979
[18] D. H. Fremlin: Topological Riesz spaces and measure theory. Cambridge Univ. Press, London-New York, 1974. · Zbl 0273.46035
[19] A. Goullet de Rugy: Representation of Banach lattices. Foundations of quantum mechanics and ordered linear spaces, LN in Physics 29, Springer-Verlag, Berlin-Heidelberg-New York, 1974, pp. 41-46. · Zbl 0286.46014
[20] W. Hackenbroch: Zur Darstellungstheorie \(\sigma \)-vollständiger Vektorverbände. Math. Z. 128 (1972), 115-128. · Zbl 0231.46017 · doi:10.1007/BF01111473
[21] W. Hackenbroch: Eindeutigkeit des Darstellungsraumes von Vektorverbänden. Math. Z. 135 (1974), 285-288. · Zbl 0263.46007 · doi:10.1007/BF01215367
[22] D. G. Johnson and J. E. Kist: Complemented ideals and extremally disconnected spaces. Arch. Math. (Basel) 12 (1961), 349-354. · Zbl 0101.08701 · doi:10.1007/BF01650573
[23] D. G. Johnson and J. E. Kist: Prime ideals in vector lattices. Canad. J. Math. 14 (1962), 517-528. · Zbl 0103.33003 · doi:10.4153/CJM-1962-043-3
[24] S. Kakutani: Concrete representation of abstract \(L\)-spaces and the mean ergodic theorem. Ann. of Math. 42 (1941), 523-537. · Zbl 0027.11102 · doi:10.2307/1968915
[25] S. Kakutani: Concrete representation of abstract \(M\)-spaces. Ann. of Math. 42 (1941), 994-1024. · Zbl 0060.26604 · doi:10.2307/1968778
[26] A. V. Koldunov: Strict realization of vector lattices. Theory of functions and functional analysis, Leningrad. Gos. Ped. Inst., Leningrad, 1975, pp. 67-74. · Zbl 0755.47040
[27] M. G. Kreĭn and S. G. Kreĭn: On an inner characteristic of the set of all continuous functions defined on a bicompact Hausdorff space. Dokl. Akad. Nauk SSSR 27 (1940), 427-430. · JFM 66.0529.02
[28] M. G. Kreĭn and S. G. Kreĭn: Sur l’espace des fonctions continues définies sur un bicompact de Hausdorff et ses sous- espaces semi-ordonnés. Mat. Sb. 13(55) (1943), 1-38.
[29] I. Labuda: Submeasures and locally solid topologies on Riesz spaces. Math. Z. 195 (1987), 179-196. · Zbl 0601.46006 · doi:10.1007/BF01166456
[30] W. A. J. Luxemburg and J. J. Masterson: An extension of the concept of the order dual of a Riesz space. Canad. J. Math. 19 (1967), 488-498. · Zbl 0147.11101 · doi:10.4153/CJM-1967-041-6
[31] W. A. J. Luxemburg and A. C. Zaanen: Riesz spaces I. North-Holland Publ. Comp., Amsterdam -London, 1971. · Zbl 0231.46014
[32] F. Maeda and T. Ogasawara: Representation of vector lattices. J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17-35.
[33] B. M. Makarov and M. Weber: On the realization of vector lattices I. Math. Nachr. 60 (1974), 281-296. · Zbl 0364.46006
[34] J. T. Marti: Topological representation of abstract \(L_p\)-spaces. Math. Ann. 185 (1970), 315-321. · Zbl 0182.45501 · doi:10.1007/BF01349954
[35] H. Nakano: Eine Spektraltheorie. Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 485-511. · Zbl 0060.26504
[36] T. Ogasawara: Remarks on the representation of vector lattices. J. Sci. Hiroshima Univ. Ser. A 12 (1943), 217-234.
[37] A. G. Pinsker: On concrete representations of linear semi-ordered spaces. C. R. (Doklady) Acad. Sci. URSS (N. S.) 55 (1947), 379-381. · Zbl 0029.04903
[38] H. H. Schaefer: On the representation of Banach lattices by continuous numerical functions. Math. Z. 125 (1972), 215-232. · Zbl 0216.40702 · doi:10.1007/BF01111305
[39] H. H. Schaefer: Banach lattices and positive operators. Springer-Verlag, New York-Heidelberg-Berlin, 1974. · Zbl 0296.47023
[40] Z. Semadeni: Banach spaces of continuous functions I. Polish Scientific Publ., Warsaw, 1971. · Zbl 0225.46030
[41] A. I. Veksler: Realizations of Archimedean \(K\)-lineals. Sibirsk. Mat. Zh. 3 (1962), 7-16. · Zbl 0107.08603
[42] A. I. Veksler: Localness of functional vector lattices. Siberian Math. J. 12 (1971), 39-46. · Zbl 0255.46005 · doi:10.1007/BF00969139
[43] A. I. Veksler: The Kreĭn-Kakutani Theorem and realization characteristics of \(K_{\sigma }\)-spaces. Functional analysis, No. 9: Harmonic analysis on groups, Ul’janovsk. Gos. Ped. Inst., Ul’yanovsk, 1977, pp. 8-16. · Zbl 0408.46010
[44] V. K. Vietsch: Abstract kernel operators and compact operators. Ph. D. thesis, Leiden Univ., 1979.
[45] B. Z. Vulikh: Concrete representations of linear partially ordered spaces. Dokl. Akad. Nauk SSSR 58 (1947), 733-736.
[46] B. Z. Vulikh and G. Ya. Lozanovskiĭ: On the representation of completely linear and regular functionals in partially ordered spaces. Math. USSR-Sb. 13 (1971), 323-343. · Zbl 0238.47024 · doi:10.1070/SM1971v013n03ABEH003686
[47] M. Weber: On the realization of vector lattices on locally compact topological spaces. Proc. Conf. Topology and Measure I, Wissensch. Beitr. Ernst-Moritz-Arndt-Univ. Greifswald, 1978, pp. 393-401. · Zbl 0434.06017
[48] M. Wolff: Darstellung von Banach-Verbänden und Sätze vom Korovkin-Typ. Math. Ann. 200 (1973), 47-67. · Zbl 0236.46010 · doi:10.1007/BF01578291
[49] W. Wnuk: Representations of Orlicz lattices. Diss. Math. 235 (1984). · Zbl 0566.46018
[50] K. Yosida: On the representation of the vector lattice. Proc. Imp. Acad. Tokyo 18 (1941-42), 339-342. · Zbl 0063.09070 · doi:10.3792/pia/1195573861
[51] A. C. Zaanen: Riesz spaces II. North-Holland Publ. Comp., Amsterdam-New York-London, 1983. · Zbl 0519.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.