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Integral representation theorems in partially ordered vector spaces

Part of: Measures, integration, derivative, holomorphy Set functions, measures and integrals with values in abstract spaces

Published online by Cambridge University Press:  09 April 2009

Panaiotis K. Pavlakos
Panaiotis K. Pavlakos
Affiliation:
Department of Mathematics University of AthensAthens, Greece
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Abstract

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Defining a Radon-type integration process we extend the Alexandroff, Fichtengolts-KantorovichHildebrandt and Riesz integral representation theorems in partially ordered vector spaces.

We also identify some classes of operators with other classes of operator-valued set functions, the correspondence between operator and operator-valued set function being given by integration.

All these established results can be immediately applied in C* -algebras (especially in W* -algebras and AW* -algebras of type I), in Jordan algebras, in partially ordered involutory (O*-)algebras, in semifields, in quantum probability theory, as well as in the operator Feynman-Kac formula.

MSC classification

Secondary: 28B15: Set functions, measures and integrals with values in ordered spaces 28B05: Vector-valued set functions, measures and integrals 46G10: Vector-valued measures and integration
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 187 - 215
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Alfsen, E. M., Compact convex sets and boundary integrals, Springer-Verlag, 1971.CrossRefGoogle Scholar
[2]Alfsen, E. M., Shultz, F. W. and Størmer, E., ‘A Gelfand-Neumark theorem for Jordan algebras, Adv. in Math. 28 (1978), 11–56.CrossRefGoogle Scholar
[3]Antonovskii, N. Ja., Boltjanskii, V. G. and Sarymsakov, T. A., Topological Boolean algebra, Izdat. Akad. Nauk Uzbek. SSR, Tashkent, 1963Google Scholar
(Russian). English translation, ‘Topological semifields and their application to general topology’, Amer. Math. Soc. Transl. 106 (1977).Google Scholar
[4]Alexandroff, A. D., ‘Additive set functions in abstract spaces’, Rec. Math. (Mat. Sbornik) N.S. (2) 51 (1941), 563–621.Google Scholar
[5]Bartle, R. G., Dunford, N. and Schwartz, J., ‘Weak compactness and vector measures’, Canad. J. Math. 7 (1955), 289–305.CrossRefGoogle Scholar
[6]Bartle, R. G., ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337–352.CrossRefGoogle Scholar
[7]Batt, J. and Berg, E. J., ‘Linear bounded transformations on the space of continuous functions’, J. Funct. Anal. 4 (1969), 215–239.CrossRefGoogle Scholar
[8]Berberian, S. K., Notes on spectral theory, Van Nostrand, Princeton, N.J., 1966.Google Scholar
[9]Bochner, S., ‘Integration and differentiation in partially ordered spaces’, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 29–31.CrossRefGoogle ScholarPubMed
[10]Bonsall, F. F. and Duncan, J., Complete normed algebras, Springer-Verlag, 1973.CrossRefGoogle Scholar
[11]Bourbaki, N., Integration, 2nd ed., Chapters 1–4, Hermann, Paris, 1965, Chapters 7 and 8, 1963.Google Scholar
[12]Brooks, J. K. and Lewis, P. W., ‘Linear operators and vector measures’, Trans. Amer. Math. Soc. 192 (1974), 139–162.CrossRefGoogle Scholar
[13]Brooks, J. K. and Dinculeanu, N., ‘Lebesgue-type spaces for vector integration, Linear operators weak completeness and weak compactness’, J. Math. Anal. Appl. 54 (1976), 348–389.CrossRefGoogle Scholar
[14]Cattaneo, U., ‘On Mackey's imprimitivity theorem’, Comment. Math. Helv. 54 (1979), 629–641.CrossRefGoogle Scholar
[15]Christian, R. R., ‘On order-preserving integration’, Trans. Amer. Math. Soc. 86 (1957), 463–488.CrossRefGoogle Scholar
[16]Cristescu, R., ‘Integrali vettoriali su uno spazio localmente compatto’, Rend. Accad. Naz. Lincei, Roma Ser. VIII, 40 (1966), 792–795.Google Scholar
[17]Cristescu, R., Ordered spaces and linear operators, Bucuresti Ed. Academiei, Abacus Press, Kent, England, 1976.Google Scholar
[18]Davies, E. B., ‘On the Borel structure of C* -algebras’, Comm. Math. Phys. 8 (1968), 147–163.CrossRefGoogle Scholar
[19]Davies, E. B., ‘On the repeated measurement of continuous observables in quantum mechanics’, J. Funct. Anal. 6 (1970), 318–346.CrossRefGoogle Scholar
[20]Davies, E. B. and Lewis, J. T., ‘An operational approach to quantum probability’, Comm. Math. Phys. 17 (1970), 239–260.CrossRefGoogle Scholar
[21]Diestel, J. J. and Uhl, J. J. Jr, Vector measures, Math. Surveys, No. 15, Amer. Math. Soc., Providence, R. I., 1977.CrossRefGoogle Scholar
[22]Dinculeanu, N., Vector measures, Pergamon Press, New York, 1967.CrossRefGoogle Scholar
[23]Dixmier, J., Les algèbres d'opérateurs dans l'espace Hilbertien, 2ième éd., Gauthier-Villars, Paris, 1969.Google Scholar
[24]Dixmier, J., Les C*-algèbres et leurs représentations, 2ième éd., Gauthier-Villars, Paris, 1969.Google Scholar
[25]Dobrakov, I., ‘On integration in Banach spaces. I, II’, Czechoslovak Math. J. 20 (95) (1970), 511–536, 680–695.CrossRefGoogle Scholar
[26]Dobrakov, I., ‘On representation of linear operators on C0(T, X)’, Czechoslovak Math. J. 21 (1971), 13–30.CrossRefGoogle Scholar
[27]Dunford, N., ‘A bilinear integral I’, Adv. in Math. 17 (1975), 337–342.CrossRefGoogle Scholar
[28]Easton, R. J. and Tucker, D. H., ‘A generalized Lebesgue-type integral’, Math. Ann. 181 (1969), 311–324.CrossRefGoogle Scholar
[29]Edwards, C. M., ‘The operational approach to algebraic quantum theory I’, Comm. Math. Phys. 16 (1970), 207–230.CrossRefGoogle Scholar
[30]Edwards, C. M., ‘Classes of operations in quantum theory’, Comm. Math. Phys. 20 (1971), 26–56.CrossRefGoogle Scholar
[31]Edwards, J. R. and Wayment, S. G., ‘A unifying representation theorem’, Math. Ann. 187 (1970), 317–328.CrossRefGoogle Scholar
[32]Effros, E. G. and Størmer, E., ‘Jordan algebras of self-adjoint operators’, Trans. Amer. Math. Soc. 127 (1967), 313–316.CrossRefGoogle Scholar
[33]Fichtengolts, G. M. and Kantorovich, L. V., ‘Sur les opérations linéaires dans l'espace des fonctions’, Studia Math. 5 (1935), 70–98.Google Scholar
[34]Goldstein, J. A., Semigroups of linear operators and applications, Oxford Univ. Press, Clarendon Press, 1985.Google Scholar
[35]Goodrich, R. K., ‘A Riesz representation theorem’, Proc. Amer. Math. Soc. 24 (1970), 629–636.CrossRefGoogle Scholar
[36]Hartkamper, A. and Neumann, H. (eds), Foundations of quantum mechanics and ordered linear spaces, Lecture Notes in Physics, vol. 29, Springer-Verlag, 1974.CrossRefGoogle Scholar
[37]Kantorovich, L. V., Vulikh, B. Z. and Pinsker, A. G., Functional analysis in partially ordered spaces, Gostekhizdat, 1950 (Russian).Google Scholar
[38]Kantorovich, L. V. and Vulikh, B. Z., ‘Sur la représentation des opérations linéaires’, Compositio Math. 5 (1937), 119–165.Google Scholar
[39]Kappos, D. A., Probability algebras and stochastic spaces, Academic Press, New York and London, 1969.Google Scholar
[40]Kelley, J. L., Namioka, I. and co-authors, Linear topological spaces, Van Nostrand, Princeton, N. J., 1963.CrossRefGoogle Scholar
[41]Khurana, S. S., ‘Lattice-valued Borel measures’, Rocky Mountain J. Math. 6 (1976), 377–382.CrossRefGoogle Scholar
[42]Kluvanek, I., ‘Operator valued measures and perturbations of semigroups’, Arch. Rational Mech. Anal. 81 (1983), 161–180.CrossRefGoogle Scholar
[43]Kluvanek, I., ‘Integration and the Feynman-Kac formula’, Studia Math. 87 (1987), 35–57.CrossRefGoogle Scholar
[44]Kuckarov, Ja. H., ‘Convergence of semifield-valued measures in the Prohorov metric’, Dokl. Akad. Nauk SSSR 237 (2) (1977), 260–263Google Scholar
(Russian). English translation, Soviet Math. Dokl. 18 (6) (1977), 1415–1418.Google Scholar
[45]Kucharov, Ja. H., ‘Convergence of semifield-valued distributions and a generalization of the Lindeberg-Feller theorem’, Dokl. Akad. Nauk SSSR 237 (6) (1977), 1273–1276Google Scholar
(Russian). English translation, Soviet Math. Dokl. 18 (6) (1977), 1540–1544.Google Scholar
[46]Mazur, S. and Orlicz, W., ‘Sur les espaces métriques linéaires. I, II’, Studia Math. 10 (1948), 184–208,CrossRefGoogle Scholar
‘Sur les espaces métriques linéaires. I, II’, Studia Math. 13 (1953), 137–179.CrossRefGoogle Scholar
[47]McShane, E. J., Order-preserving maps and integration process, Ann. of Math. Studies 31, Princeton Univ. Press, Princeton, N. J., 1953.Google Scholar
[48]Morse, P. M. and Feshbach, H., Methods of theoretical physics, McGraw-Hill, New York, 1953.Google Scholar
[49]Papangelou, F., ‘Order convergence and topological completion of commutative lattice-groups’, Math. Ann. 155 (1964), 81–107.CrossRefGoogle Scholar
[50]Pavlakos, P. K., ‘On integration in partially ordered groups’, Canad. J. Math. 35 (2) (1983), 353–372.CrossRefGoogle Scholar
[51]Pavlakos, P. K., ‘Convolutions and products of partially ordered vector-valued positive measures’, Math. Ann., to appear.Google Scholar
[52]Peressini, A., Ordered topological vector spaces, Harper and Row, New York and London, 1967.Google Scholar
[53]Przeworska-Rolewicz, A. and Rolewicz, S., ‘On integrals of functions with values in a complete linear metric space’, Studia Math. 26 (1966), 121–131.CrossRefGoogle Scholar
[54]Rowecka, E., ‘Random integrals and type and cotype of Banach space’, Math. Z. 193 (1986), 381–391.CrossRefGoogle Scholar
[55]Sakai, S., C* -algebras and W* -algebras, Springer-Verlag, Berlin and New York, 1971.Google Scholar
[56]Sarymsakov, T. A., Rubštein, B. A. and Čilin, V. I., ‘Complete tensor products of topological semifields’, Dokl. Akad. Nauk SSSR 216 (6) (1974), 1226–1228Google Scholar
(Russian). English translation, Soviet Math. Dokl. 15 (3) (1974), 969–972.Google Scholar
[57]Sarymsakov, T. A., Rubštein, O. Ja. A. and Čilin, V. I., ‘Measures with values in semifields and their applications in probability theory’, Dokkl. Akad. Nauk SSSR 228 (1) (1976), 41–44Google Scholar
(Russian). English translation, Soviet Math. Dokl. 17 (3) (1976), 656–659.Google Scholar
[58]Sarymsakov, T. A., Semifields and probability theory, Lecture Notes in Math., vol. 550, Springer-Verlag, Berlin and New York, 1976, pp. 525–549.Google Scholar
[59]Sarymsakov, T. A., and Gol'dštein, M. Š., ‘On partially ordered involutory algebras’, Dokl. Akad. Nauk SSSR 228 (2) (1976), 306–309Google Scholar
(Russian). English translation, Soviet Math. Dokl. 17 (3) (1976), 725–729.Google Scholar
[60]Sarymsakov, T. A., Kučkarov, Ja. H. and Dauletbaev, T. E., ‘Convergence of Boolean measures in metric spaces’, Dokl. Akad. Nauk SSSR 232 (5) (1977), 1023–1025Google Scholar
(Russian). English translation, Soviet Math. Dokl. 18 (1) (1977), 206–208.Google Scholar
[61]Sarymsakov, T. A., ‘Noncommutative probability spaces on O* -algebras’, Dokl. Akad. Nauk SSSR 241 (2) (1978), 297–300Google Scholar
(Russian). English translation, Soviet Math. Dokl. 19 (4) (1978), 855–858.Google Scholar
[62]Sarymsakov, T. A. and Halikulov, I. B., ‘On the strong law of large numbers relative to semifield-valued measures’, Dokl. Akad. Nauk SSSR 238 (4) (1978), 808–810Google Scholar
(Russian). English translation, Soviet Math. Dokl. 19 (1) (1978), 116–118.Google Scholar
[63]Schaefer, H. H., Topological vector spaces, Macmillan, New York, 1966.Google Scholar
[64]Shuchat, A. H., ‘Integral representation theorems in topological vector spaces’, Trans. Amer. Math. Soc. 172 (1972), 373–397.CrossRefGoogle Scholar
[65]Størmer, E., ‘On the Jordan structure of C* -algebra’, Trans. Amer. Math. Soc. 120 (1965), 438–447.Google Scholar
[66]Størmer, E., ‘Jordan algebras of type I’, Acta Math. 115 (1966), 165–184.CrossRefGoogle Scholar
[67]Størmer, E., ‘Irreducible Jordan algebras of self-adjoint operators’, Trans. Amer. Math. Soc. 130 (1968), 153–166.CrossRefGoogle Scholar
[68]Swong, K., ‘A representation theory of continuous linear maps’, Math. Ann. 155 (1964), 270–291;CrossRefGoogle Scholar
errata, Math. Ann. 157 (1964), 178.Google Scholar
[69]Tanabe, H., Equations of evolution, Pitman Press, 1979.Google Scholar
[70]Topping, D., ‘Jordan algebras of self-adjoint operators’, Mem. Amer. Math. Soc. 53 (1965).Google Scholar
[71]Tsitsas, L. N., ‘Integral representations of linear maps’, Bull. Soc. Math. Gréce (N.S.) 6 II, fasc. 2 (1965), 298–355.Google Scholar
[72]Uherka, D. H., ‘Generalized Stieltjes integrals and a strong representation theorem for continuous linear maps on a function space’, Math. Ann. 182 (1969), 60–66.CrossRefGoogle Scholar
[73]Umegaki, H. and Bharucha-Reid, A. T., ‘Banach space-valued random variables and tensor products of Banach spaces’, J. Math. Anal. Appl. 31 (1970), 49–67.CrossRefGoogle Scholar
[74]Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Fizmatgiz, Moscow, 1961). English translation, Noordhoff, Groningen, 1967.Google Scholar
[75]Wickstead, A. W., ‘Stone-algebra-valued measures: Integration of vector-valued functions and Radon-Nikodym type theorems’, Proc. London Math. Soc. 45 (3) (1982), 193–226.CrossRefGoogle Scholar
[76]Wright, J. D. M., ‘Stone-algebra-valued measures and integrals’, Proc. London. Math. Soc. 19 (1969), 107–122.CrossRefGoogle Scholar
[77]Wright, J. D. M., ‘Vector-lattice measures on locally compact spaces’, Math. Z. 120 (1971), 193–203.CrossRefGoogle Scholar
[78]Wright, J. D. M., ‘Measures with values in a partially ordered vector space’, Proc. London Math. Soc. 25 (1972), 655–688.Google Scholar
[79]Wright, J. D. M., ‘Products of positive vector measures’, Quart. J. Math. (2) 24 (1973), 189–206.CrossRefGoogle Scholar
[80]Wright, J. D. M., ‘On minimal σ-completions of C* -algebras’, Bull. London Math. Soc. 6 (1974), 168–174.CrossRefGoogle Scholar
[81]Wright, J. D. M., ‘Regular σ-completions of C* -algebras’, J. London Math. Soc. 45 (2) (1976), 299–309.CrossRefGoogle Scholar
[82]Wright, J. D. M., ‘Jordan C* -algebras’, Michigan Math. J. 24 (1977), 291–302.CrossRefGoogle Scholar
[83]Yosida, K., Functional analysis, 6th ed., Springer-Verlag, Berlin and New York, 1980.Google Scholar