Abstract
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
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References
Y.A. Abramovich, A.I. Veksler, A.V. Koldunov, On operators preserving disjointness, Soviet Math. Dokl. 20 (1979), pp. 1089–1093.
S.J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15 (1965), pp. 599–631.
S.J. Bernau, A note on 4-spaces, Math. Ann. 200 (1973), pp. 281–286.
S.J. Bernau, Orthomorphisms of Archimedean vector lattices, Proc. Camb. Phil. Soc. 89 (1979), pp. 119–128.
H.F. Bohnenblust, On axiomatic characterisation of Lp spaces, Duke Math. J. 6 (1940), pp. 627–640.
H.F. Bohnenblust, S. Kakutani, Concrete representations of (M)-spaces, Ann. Math. 42 (1941), pp. 1025–1028.
J. Bretagnolle, D. Dacunha-Castelle, J.L. Krivine, Lois stables et espaces LP, Ann. Inst. H. Poincare 2 (1965/6), pp. 231–259.
E.B. Davies, The structure and ideal theory of the predual of a Banach lattice, Trans. Amer. Math. Soc. 131 (1968), pp. 544–555.
E.B. Davies, The Choquet theory and representation of ordered Banach spaces, Illinois J. Math. 13 (1969), pp. 176–187.
D.H. Fremlin, Abstract Kothe spaces II, Proc. Cam. Phil. Soc. 63 (1967), pp. 951–956.
H. Gordon, Measures defined by abstract L p spaces, Pacific J. Math. 10 (1960), pp. 557–562.
A. Goullet de Rugy, La théorie des cônes biréticulés, Ann. Inst. Fourier (Grenoble) 21 (1971), pp. 1–18.
A. Goullet de Rugy, La structure ideale des M-espaces, J. Maths. Pures et Appl. 51 (1972), pp. 331–373.
R. Haydon, Injective Banach lattices, Math. Z. 156 (1977), pp. 19–47.
S. Kakutani, Concrete representation of abstract L-spaces and the mean ergodic theorem, Ann. Math. 42 (1941), pp. 523–537.
S. Kakutani, Concrete representations of abstract M-spaces, Ann. Math. 42 (1941), pp. 994–1024.
L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, Functional Analysis in Partially Ordered Spaces (Russian), Gostekhizdat, Moscow, 1950.
M. Klein, 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.
H.E. Lacey, S.J. Bernau, Characterisations and classifications of some classical Banach spaces, Ade. in Math. 12 (1974), pp. 367–401.
H.P. Lotz, Zur Idealstruktur in Banachverbänden, Habilitationsschrii t Tubingen (1969).
W. A. J. Luxemburg, A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam-London, 1971.
W.A.J. Luxemburg, Some Aspects of the Theory of Riesz Spaces, Univ. of Arkansas Lecture Notes, Fayetteville (1979).
F. Maeda, T. Ogasawara, Representation of vector lattices, J. Sci. Hiroshima Univ. 12 (1942), pp. 17–35.
J.T. Marti, Topological representations of abstract Lp spaces, Math. Ann. 185 (1970), pp. 315–321.
P.T.N. McPolin, Disjointness preserving linear mappings on a vector lattice, Ph.D. Thesis, Q.U.B. (1983).
P.T.N. McPolin, A.W. Wickstead, The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Pmc. Cam. Phil. Soc. 97 (1985), pp. 481–487.
M. Meyer, Representations des espaces vectoriels réticulés, Seminaire Chuquet 13 (1973/4), pp. 1–12.
M. Meyer, Quelques propriétés des homomorphismes d’espaces vectoriels réticulés, E.R.A. Université Paris VI 294 (1978).
R.J. Nagel, Darstellung von Verbandsoperatoren auf Banach- verbänden, Rev. Acad. Ci. Zaragoza 27 (1972), pp. 281–288.
R.J. Nagel, Ordnungstetigkeit in Banachverbänden, Manuscripta Math 9 (1973), pp. 9–27.
R.J. Nagel, A Stone-Weierstrass theorem for Banach lattices, Studia Math. 47 (1973), pp. 75–82.
H. Nakano, Über die Charakterisierung des allgemeinen C-Raumes, Proc. Imp. Acad. Tokyo 17 (1941), pp. 301–307.
H. Nakano, Über die Charakterisierung des allgemeinen C-Raumes II, Proc. Imp. Acad. Tokyo 18 (1942), pp. 280–286.
H. Nakano, Stetige lineare Funktionale auf dem teilweisgeordneten Modul, J. Fac. Sci. Imp. Univ. Tokyo 4 (1942), pp. 201–382.
H. Nakano, Über das System aller stetiger Funktionen auf ein topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), pp. 308–310.
T. Ogasawara, Theory of vector lattices I, J. Sci. Hiroshima Univ. 12 (1942), pp. 37–100.
T. Ogasawara, Theory of vector lattices II, J. Sci. Hiroshima Univ. 13 (1944), pp. 41–161.
H.H. Schaefer, On the representation of Bauach lattices by continuous numerical functions, Math. Z. 125 (1972), pp. 215–232.
H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag. BerlinHeidelberg-New York, 1974.
B.Z. Vulikh, Concrete representation of partially ordered linear spaces (Russian), Doklady Akad. Nauk USSR 58 (1947), pp. 733–736.
B.Z. Vulikh, On concrete representation of partially ordered lineals (Russian), Doklady Akad. Nauk USSR 78 (1951), pp. 189–192.
B.Z. Vulikh, Some topics in the theory of partially ordered linear spaces (Russian), Izvestia AN USSR see. math. 17 (1953), pp. 365–388.
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.
A.W. Wickstead, Representation and duality of multiplication operators on Archimedean Riesz spaces, Compositio Math. 35 (1977), pp. 225–238.
K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 18 (1941/2), pp. 339–342.
A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam-New York-Oxford, 1983.
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Wickstead, A.W. (1992). Representations of Archimedean Riesz Spaces by Continuous Functions. In: Huijsmans, C.B., Luxemburg, W.A.J. (eds) Positive Operators and Semigroups on Banach Lattices. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2721-1_13
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DOI: https://doi.org/10.1007/978-94-017-2721-1_13
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