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Representations of Archimedean Riesz spaces by continuous functions. (English) Zbl 0802.46019

Summary: The author gives a brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces.

MSC:

46A40 Ordered topological linear spaces, vector lattices
46B40 Ordered normed spaces
46E05 Lattices of continuous, differentiable or analytic functions
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[1] Y.A. Abramovich, A.I. Veksler, A.V. Koldunov, On operators preserving disjointness, Soviet Math. Dokl. 20 (1979), pp. 1089-1093. · Zbl 0445.46017
[2] S.J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15 (1965), pp. 599-631. · Zbl 0134.10802 · doi:10.1112/plms/s3-15.1.599
[3] S.J. Bernau, A note on L p -spaces, Math. Ann. 200 (1973), pp. 281-286. · Zbl 0242.46025 · doi:10.1007/BF01428259
[4] S.J. Bernau, Orthomorphisms of Archimedean vector lattices, Proc. Camb. Phil. Soc. 89 (1979), pp. 119-128. · Zbl 0463.46002 · doi:10.1017/S030500410005800X
[5] H.F. Bohnenblust, On axiomatic characterisation of L p spaces, Duke Math. J. 6 (1940), pp. 627-640. · JFM 66.0537.05 · doi:10.1215/S0012-7094-40-00648-2
[6] H.F. Bohnenblust, S. Kakutani, Concrete representations of (M)-spaces, Ann. Math. 42 (1941), pp. 1025-1028. · Zbl 0061.24209 · doi:10.2307/1968779
[7] J. Bretagnolle, D. Dacunha-Castelle, J.L. Krivine, Lois stables et espaces L p , Ann. Inst. H. Poincare 2 (1965/6), pp. 231-259. · Zbl 0139.33501
[8] E.B. Davies, The structure and ideal theory of the predual of a Banach lattice, Trans. Amer. Math. Soc. 131 (1968), pp. 544-555. · Zbl 0159.41503 · doi:10.1090/S0002-9947-1968-0222604-8
[9] E.B. Davies, The Choquet theory and representation of ordered Banach spaces, Illinois J. Math. 13 (1969), pp. 176-187. · Zbl 0165.46801
[10] D.H. Fremlin, Abstract Kothe spaces II, Proc. Cam. Phil. Soc. 63 (1967), pp 951-956. · Zbl 0179.17005 · doi:10.1017/S0305004100041979
[11] H. Gordon, Measures defined by abstract L p spaces, Pacific J. Math. 10 (1960), pp. 557-562. · Zbl 0178.49905
[12] A. Goullet de Rugy, La théorie des cônes biréticulés, Ann. Inst. Fourier (Grenoble) 21 (1971), pp. 1-18. · Zbl 0215.48001
[13] A. Goullet de Rugy, La structure ideale des M-espaces J. Maths. Pures et Appl. 51 (1972), pp. 331-373.
[14] R. Haydon, Injective Banach lattices, Math. Z 156 (1977), pp. 19-47. · Zbl 0353.46003 · doi:10.1007/BF01215126
[15] S. Kakutani, Concrete representation of abstract L-spaces and the mean ergodic theorem, Ann. Math. 42 (1941), pp. 523-537. · Zbl 0027.11102 · doi:10.2307/1968915
[16] S. Kakutani, Concrete representations of abstract M-spaces, Ann. Math. 42 (1941), pp. 994-1024. · Zbl 0060.26604 · doi:10.2307/1968778
[17] L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, Functional Analysis in Partially Ordered Spaces (Russian), Gostekhizdat, Moscow, 1950. · Zbl 0037.07201
[18] M. Krein, S. Krein, On an inner characterisation of the set of all continuous functions defined on a bicompact Hausdorff space, C.R. Acad. Sci. URSS 27 (1940), pp. 427-430. · Zbl 0023.32701
[19] H.E. Lacey, S.J. Bernau, Characterisations and classifications of some classical Banach spaces, Adv. in Math. 12 (1974), pp. 367-401. · Zbl 0278.46027 · doi:10.1016/S0001-8708(74)80008-3
[20] H.P. Lotz, Zur Idealstruktur in Banachverbänden, Habilitationschrift Tubingen (1969).
[21] W. A. J. Luxemburg, A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam-London, 1971.
[22] W.A.J. Luxemburg, Some Aspects of the Theory of Riesz Spaces, Univ. of Arkansas Lecture Notes, Fayetteville (1979). · Zbl 0431.46003
[23] F. Maeda, T. Ogasawara, Representation of vector lattices, J. Sci. Hiroshima Univ. 12 (1942), pp. 17-35.
[24] J.T. Marti, Topological representations of abstract L p -spaces, Math. Ann. 185 (1970), pp. 315-321. · Zbl 0187.05201 · doi:10.1007/BF01349954
[25] P.T.N. McPolin, Disjointness preserving linear mappings on a vector lattice, Ph.D. Thesis, Q.U.B. (1983).
[26] P.T.N. McPolin, A.W. Wickstead, The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Proc. Cam. Phil. Soc. 97 (1985), pp. 481-487. · Zbl 0564.47017 · doi:10.1017/S0305004100063052
[27] M. Meyer, Representations des espaces vectoriels réticulés Seminaire Choquet 13 (1973/4), pp. 1-12.
[28] M. Meyer, Quelques propriétés des homomorphismes d’espaces vectoriels réticulés, E.R.A. Université Paris VI 294 (1978).
[29] R.J. Nagel Darstellung von Verbandsoperatoren auf Banachverbänden, Rev. Acad. Ci. Zaragoza 27 (1972), pp. 281-288.
[30] R.J. Nagel, Ordnungstetigkeit in Banachverbänden, Manuscripta Math 9 (1973), pp. 9-27. · Zbl 0248.46010 · doi:10.1007/BF01320666
[31] R.J. Nagel, A Stone-Weierstrass theorem for Banach lattices, Studia Math. 47 (1973), pp. 75-82. · Zbl 0255.46008
[32] H. Nakano, Über die Charakterisierung des allgemeinen C-Raumes, Proc. Imp. Acad. Tokyo 17 (1941), pp. 301-307. · Zbl 0060.26509 · doi:10.3792/pia/1195578668
[33] H. Nakano, Über die Charakterisierung des allgemeinen C-Raumes II, Proc. Imp. Acad. Tokyo, 18 (1942), pp. 280-286. · Zbl 0060.26509 · doi:10.3792/pia/1195573902
[34] H. Nakano, Stetige lineare Funktionale auf dem teilweisgeordneten Modul, J. Fac. Sci. Imp. Univ. Tokyo 4 (1942), pp. 201-382. · Zbl 0060.26506
[35] H. Nakano, Über das System aller stetiger Funktionen auf ein topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), pp. 308-310. · Zbl 0060.26508 · doi:10.3792/pia/1195578669
[36] T. Ogasawara, Theory of vector lattices I, J. Sci. Hiroshima Univ. 12 (1942), pp. 37-100. · Zbl 0063.06000
[37] T. Ogasawara, Theory of vector lattices II, J. Sci. Hiroshima Univ. 13 (1944), pp. 41-161. · Zbl 0063.06001
[38] H.H. Schaefer, On the representation of Banach lattices by continuous numerical functions, Math. Z. 125 (1972), pp. 215-232. · Zbl 0216.40702 · doi:10.1007/BF01111305
[39] H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1974. · Zbl 0296.47023
[40] B.Z. Vulikh, Concrete representation of partially ordered linear spaces (Russian), Doklady Akad. Nauk USSR 58 (1947), pp. 733-736.
[41] B.Z. Vulikh, On concrete representation of partially ordered lineals (Russian), Doklady Akad. Nauk USSR 78 (1951), pp. 189-192.
[42] B.Z. Vulikh, Some topics in the theory of partially ordered linear spaces (Russian), Izvestia AN USSR ser. math. 17 (1953), pp. 365-388.
[43] B.Z. Vulikh, G.Y. Lozanovskii, On representation of order continuous and regular functionals on partially ordered spaces (Russian), Mat. Sbornik 84 (1971), pp. 331-352.
[44] A.W. Wickstead, Representation and duality of multiplication operators on Archimedean Riesz spaces, Compositio Math. 35 (1977), pp. 225-238. · Zbl 0381.47021
[45] K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 18 (1941/2), pp. 339-342. · Zbl 0063.09070 · doi:10.3792/pia/1195573861
[46] A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam-New York-Oxford, 1983.
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