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Bulletin of the American Mathematical Society

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Algebraic integration theory


Author: Irving Segal
Journal: Bull. Amer. Math. Soc. 71 (1965), 419-489
DOI: https://doi.org/10.1090/S0002-9904-1965-11284-8
MathSciNet review: 0178384
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  • 1. W. Ambrose, The L. I, Trans. Amer. Math. Soc.65 (1949), 27-48. MR 28322
  • 2. S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Math. Ann. 108 (1933), no. 1, 378–410 (German). MR 1512856, https://doi.org/10.1007/BF01452844
  • 3. S. Bochner, Stochastic processes, Ann. of Math. (2) 48 (1947), 1014-1061. MR 22322
  • 4. S. Bochner, Harmonic analysis and the theory of probability, Univ. California Press, Berkeley, Calif., 1955. MR 72370
    4A. N. Bourbaki, Éléments de mathématique. XIII, Hermann, Paris, 1952. MR 580296

  • 5. R. H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623-627. MR 65033
  • 6. R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2) 45 (1944), 386-396. MR 10346
  • 7. R. H. Cameron and W. T. Martin, An expression for the solution of a class of non-linear integral equations, Amer. J. Math. 66 (1944), 281-298. MR 10050
  • 8. R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under a general class of linear transformations, Trans. Amer. Math. Soc. 58 (1945), 184-219. MR 13240
  • 9. R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of functionals belonging to L, Duke Math. J. 14 (1947), 99-107. MR 20231
  • 10. R. H. Cameron and W. T. Martin, The behavior of measure and measurability under change of scale in Wiener space, Bull. Amer. Math. Soc. 53 (1947), 130-137. MR 19259
  • 11. R. H. Cameron and W. T. Martin, The transformation of Wiener integrals by non-linear transformations, Trans. Amer. Math. Soc. 66 (1949), 253-283. MR 31196
  • 12. C. Carathéodory, Mass und Integral und ihre Algebraisierung, Birkhäuser Verlag, Basel and Stuttgart, 1956. MR 79628
    12A. J. M. Cook, The mathematics of second quantization, Trans. Amer. Math. Soc. 74 (1957), 222-245. MR 53784

  • 13. P. J. Daniell, A general form of integral, Ann. of Math. (2) 19 (1918), no. 4, 279–294. MR 1502538, https://doi.org/10.2307/1967495
  • 14. P. A. M. Dirac, The principles of quantum mechanics, 3rd ed., Oxford Univ. Press, Oxford, 1947. MR 23198
  • 15. J. Dixmier, Les fonctionnelles linéaires sur l'ensemble des opérateurs bornés d'un espace de Hilbert, Ann. of Math. (2) 51 (1950), 387-408. MR 33445
  • 16. J. Dixmier, Sur certaines espaces considérés par M. H. Stone, Summa Brasil. Math. 2 (1951), 151-182. MR 48787
  • 17. J. Dixmier, Formes linéaires sur un anneau d'opérateurs, Bull. Soc. Math. France 81 (1953), 9-39. MR 59485
    17A. J. Dixmier, Algèbres quasi-unitaires, Comment. Math. Helv. 26 (1952), 275-322. MR 52697

  • 18. J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, Gauthier-Villars, Paris, 1957 MR 94722
  • 19. H. A. Dye, The Radon-Nikodým theorem for finite rings of operators, Trans. Amer. Math. Soc. 72 (1952), 243-280. MR 45954
  • 20. J. Feldman, On the Schrödinger and heat equations for bad potentials, Trans. Amer. Math. Soc. 108 (1963), 251-264. MR 160264
  • 21. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys. 20 (1948), 367-387. MR 26940
  • 22. R. P. Feynman, An operator calculus having applications in quantum electrodynamics, Phys. Rev. (2) 84 (1951), 108-128. MR 44379
  • 23. K. O. Friedrichs, Mathematical aspects of the quantum theory of fields, Interscience, New York, 1953. MR 56468
  • 24. K. O. Friedrichs and H. N. Shapiro, Integration over Hilbert space and outer extensions, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 336-338. MR 108711
  • 25. K. O. Friedrichs and colleagues, Seminar on Integration of Functionals, Mimeographed notes published by New York University, Institute of Mathematical Sciences, 1957.
  • 26. H. Furstenberg, Stationary processes and prediction theory, Princeton Univ. Press, Princeton, N. J., 1960. MR 140151
  • 27. I. M. Gelfand, Normierte Ringe, Mat. Sb. (N.S.) 9 (51) (1941), 3-24. MR 4726
  • 28. I. M. Gelfand, Generalized random processes, Dokl. Akad. Nauk SSSR 100 (1955), 853-856. (Russian) MR 68769
  • 29. I. M. Gelfand and M. A. Naĭmark, Unitary representations of the Lorentz group, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 411-504. (Russian) MR 24440
  • 30. I. M. Gelfand and M. A. Naĭmark, Unitary representations of the classical groups, Trudy Mat. Inst. Steklov., Vol. 36, Izdat. Akad. Nauk SSSR, Moscow, 1950. (Russian) MR 46370
  • 31. I. M. Gelfand and M. I. Graev, Analogue of the Plancherel formula for the classical groups, Trudy Moskov. Mat. Obšč. 4 (1955), 375-404 (Russian); Amer. Math. Soc. Transl. (2) 9 (1958), 123-154. MR 92923
  • 32. I. M. Gelfand, D. A. Rykov, and G. E. Shilov, Commutative normed rings, Uspehi Mat. Nauk 1 (1946), 48-146 (Russian); Amer. Math. Soc. Transl. (2) 5(1957), 115-220. MR 27130
  • 33. I. M. Gelfand and N. Y. Vilenkin, Generalized functions, Vol. 4, Moscow, 1961.
  • 34. I. M. Gelfand and A. M. Yaglom, Integration in functional spaces and its applications in quantum physics, J. Mathematical Phys. 1 (1960), 48-69. MR 112604
  • 35. A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. MR 96113
  • 36. R. Godement, Sur la théorie des représentations unitaires, Ann. of Math. (2)53 (1951), 68-124. MR 38571
  • 37. R. Godement, Théorie des caractères. I. Algèbres unitaires, Ann. of Math. (2) 59 (1954), 47-62. MR 58879
  • 38. E. L. Griffin, Jr., Some contributions to the theory of rings of operators, Trans. Amer. Math. Soc. 75 (1953), 471-504; II, ibid. 79 (1955), 389-400. MR 59487
  • 39. L. Gross, Integration and non-linear transformations in Hilbert space, Trans. Amer. Math. Soc. 94 (1960), 404-440. MR 112025
  • 40. L. Gross, Measurable functions on Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 372-390. MR 147606
  • 41. L. Gross, Harmonic analysis on Hilbert space, Mem. Amer. Math. Soc. No. 46 (1963). MR 161095
  • 42. L. Gross, Classical analysis on a Hilbert space, Proc. Conf. Analysis in Function Space, Cambridge, Mass., 1964, pp. 51-68. MR 167150
  • 43. Harish-Chandra, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485-528. MR 63376
  • 44. K. Ito, Stationary random distributions, Mem. Coll. Sci. Univ. Kyoto 28 (1953), 209-223. MR 65060
  • 45. M. Kac, Probability and related topics in physical science, Proc. Summer Seminar, Boulder, Colo., 1957, Vol. I, Interscience, New York, 1959. MR 102849
  • 46. R. V. Kadison, On the additivity of the trace in finite factors, Proc. Nat. Acad Sci. U.S.A. 41 (1955), 385-387. MR 69402
  • 47. S. Kakutani, Concrete representation of abstract (L)-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523-537. MR 4095
  • 48. S. Kakutani, Determination of the spectrum of the flow of Brownian motion, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 319-323. MR 35924
    48A. S. Kakutani, Spectral analysis of stationary Gaussian processes, Proc. 4th Berkeley Sympos. Probability, Vol. 2, 1961; pp. 239-247. MR 140138

  • 49. A. Kirillov, Unitary representations of nilpotent Lie groups, Russian Math. Surveys 17 (1962), 53-104. MR 142001
  • 50. A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, 1933; pp. 1-62. MR 362415
  • 51. R. A. Kunze, L, Trans. Amer. Math. Soc. 89 (1958), 519-540. MR 100235
  • 52. C. Loewner, Grundzüge einer Inhaltlehre im Hilbertschen Raume, Ann. of Math.(2) 40 (1939), 816-833. MR 285
  • 53. L. H. Loomis, On the representation of $\sigma$-complete Boolean algebras, Bull. Amer. Math. Soc. 63 (1947), 757-760. MR 21084
  • 54. L. H. Loomis, Note on a theorem of Mackey, Duke Math. J. 19 (1952), 641-645. MR 50596
  • 55. G. W. Mackey, A theorem of Stone and von Neumann, Duke Math. J. 16 (1949), 313-326. MR 30532
  • 56. G. W. Mackey, Imprimitivity for representations of locally compact groups. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537-545. MR 31489
  • 57. G. W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193-221. MR 56611
  • 58. G. W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265-371. MR 98328
  • 59. G. W. Mackey, Point realizations of transformation groups, Illinois J. Math. 6 (1962), 327-335. MR 143874
  • 60. D. Maharam, Homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1941), 108-111. MR 6595
  • 61. D. Maharam, An algebraic characterization of measure algebras, Ann. of Math. (2) 48 (1947), 154-167. MR 18718
  • 62. F. Mautner, Unitary representations of locally compact groups. II, Ann. of Math. (2) 52 (1950), 528-556. MR 36763
  • 63. F. Mautner, On eigenfunction expansions, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 49-53. MR 52690
  • 64. E. J. McShane, Remark concerning integration, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 46-49. MR 27831
  • 65. E. J. McShane, Integrals devised for special purposes, Bull. Amer. Math. Soc. 69 (1963), 597-627. MR 162905
    65A. J. Milkman, Hermite polynomials, Hermite functionals and their integrals in real Hilbert space, Riv. Mat. Univ. Parma 6 (1955), 65-88. MR 77907

  • 66. F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, https://doi.org/10.2307/1968693
  • 67. F. J. Murray and J. von Neumann, On rings of operators. II, Trans. Amer. Math. Soc. 41 (1937), no. 2, 208–248. MR 1501899, https://doi.org/10.1090/S0002-9947-1937-1501899-4
  • 68. F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716-808. MR 9096
  • 69. M. Nakamura and H. Umegaki, On a proposition of von Neumann, Kodai Math. Sem. Rep. 8 (1956), 142-144. MR 90790
  • 70. H. Nakano, Hilbert algebras, Tôhoku Math. J. 2 (1950), 4-23. MR 41362
  • 71. Edward Nelson, Feynman integrals and the Schrödinger equation, J. Mathematical Phys. 5 (1964), 332-343. MR 161189
  • 72. Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142
  • 73. R. Pallu de la Barrière, Algèbres unitaires et espaces d'Ambrose, Ann. Sci. École Norm. Sup. (3) 70 (1953), 381-401. MR 60748
    73A. R. Pallu de la Barrière, Sur les algèbres d'opérateurs dans les espaces hilbertiens, Bull. Soc. Math. France 82 (1954), 1-52. MR 65822

  • 74. M. Plancherel, Contribution à l'étude de la représentation d'une fonction arbitraire par des integrals définie, Rend. Circ. Mat. Palermo 30 (1910), 289-335.
  • 75. A. Plessner, Eine Kennzeichnung der total-stetigen Funktionen, J. Reine Angew. Math. 160 (1929), 26-32.
  • 76. Y. V. Prohorov, The method of characteristic functionals, Proc. 4th Berkeley Sympos. Probability, Vol. 2, 1961; pp. 403-420. MR 133846
  • 77. L. Pukanszky, The theorem of Radon-Nikodým in operator-rings, Acta Sci. Math. (Szeged) 15 (1954), 149-156. MR 60157
    77A. V. A. Rokhlin, Unitary rings, Dokl. Akad. Nauk SSSR 59 (1948), 643-646. MR 24057

  • 78. Proceedings of the Symposium on Time Series Analysis, M. Rosenblatt, editor, pp. 279-348, Wiley, New York, 1963. MR 145634
  • 79. D. A. Rykov, Harmonic analysis on commutative groups, Trudy Mat. Inst. Steklov. 14 (1945).
  • 80. S. Saks, Theory of the integral, 2nd ed.; Note II by S. Banach, Lebesgue integration in abstract spaces, pp. 320-330. MR 167578
  • 81. V. Sazonov, On characteristic functionals, Teor. Verojatnost. i Primenen. 3 (1958), 201-205. MR 98423
  • 82. R. Schatten, The space of completely continuous operators on a Hilbert space, Math. Ann. 134 (1957), 47-49. MR 90021
  • 83. I. Segal, Postulates for general quantum mechanics, Ann. of Math. (2) 48 (1947), 930-948. MR 22652
  • 84. I. Segal, A kind of abstract integration pertinent to locally compact groups. I, Abstract, Bull. Amer. Math. Soc. 55 (1949), 46.
  • 85. I. Segal, The two-sided regular representation of a unimodular locally compact group, Ann. of Math. (2) 51 (1950), 293-298. MR 36764
  • 86. I. Segal, An extension of Plancherel's formula to separable unimodular groups, Ann. of Math. (2) 52 (1950), 272-292. MR 36765
  • 87. I. Segal, Decompositions of operator algebras. I, II. Multiplicity theory, Mem. Amer. Math. Soc. No. 9, (1951). MR 44749
  • 88. I. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-457. MR 54864
  • 89. I. Segal, Abstract probability spaces and a theorem of Kolmogoroff, Amer. J. Math. 76 (1954), 721-732. MR 63602
  • 90. I. Segal, Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81 (1956), 106-134. MR 76317
  • 91. I. Segal, Tensor algebras over Hilbert spaces. II, Ann. of Math. (2) 63 (1956), 160-175. MR 77908
  • 92. I. Segal, Ergodic subgroups of the orthogonal group on a real Hilbert space, Ann. of Math. (2) 66 (1957), 297-303. MR 89382
  • 93. I. Segal, Equivalences of measure spaces, Amer. J. Math. 73 (1951), 275-313. MR 41191
  • 94. I. Segal, Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc. 88 (1958), 12-41. MR 102759
  • 95. I. Segal, A theorem on the measurability of group-invariant operators, Duke Math. J. 26 (1959), 549-552. MR 110955
    95A. I. Segal, Foundations of the theory of dynamical systems of infinitely many degrees of freedom. I, Mat. Fys. Medd. Danske Vid. Selsk. 31 (1959), no. 12. MR 112626

  • 96. I. Segal, Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand. 13 (1963), 31-43. MR 163183
  • 97. I. Segal, Mathematical problems of relativistic physics, Lectures in Applied Mathematics, Amer. Math. Soc., Providence, R. I., 1963. MR 144227
  • 98. T. I. Seidman, Linear transformations of a functional integral, Comm. Pure Appl. Math. 12 (1959), 611-621; correction, ibid. 16 (1963), 95-96; II, ibid. 17(1964), 493-508. MR 110009
  • 99. D. Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc.103 (1962), 149-167. MR 137504
  • 100. D. Shale and W. F. Stinespring, States of the Clifford algebra, Ann. of Math.(2) 80 (1964), 365-381. MR 165880
  • 101. W. F. Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), 15-56. MR 102761
  • 102. Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877
    M. H. Stone, On one-parameter unitary groups in Hilbert space, Ann. of Math. (2) 33 (1932), no. 3, 643–648. MR 1503079, https://doi.org/10.2307/1968538
  • 103. M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), no. 1, 37–111. MR 1501865, https://doi.org/10.1090/S0002-9947-1936-1501865-8
  • 104. M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481. MR 1501905, https://doi.org/10.1090/S0002-9947-1937-1501905-7
  • 105. M. H. Stone, Notes on integration. I, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 336-342; II, ibid. 34 (1948), 447-455; III, ibid. 34 (1948), 483-490; IV, ibid. 35(1949), 50-58. MR 25552
  • 106. M. H. Stone, A general theory of spectra. I, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 280-283; II, ibid. 27 (1941), 83-87. MR 2023
  • 107. J. v. Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), no. 1, 570–578 (German). MR 1512685, https://doi.org/10.1007/BF01457956
  • 108. J. von Neumann, Über Funktionen von Funktionaloperatoren, Ann. of Math. (2) 32 (1931), no. 2, 191–226 (German). MR 1502991, https://doi.org/10.2307/1968185
  • 109. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932. MR 223138
    109A. J. von Neumann, Approximative properties of matrices of high order, Portugal. Math. 3 (1942), 1-62. MR 6137

  • 110. J. von Neumann, On an algebraical generalization of the quantum mechanical formalism. I, Mat. Sb. (N.S.) 1 (437) (1936), 415-484.
  • 111. J. von Neumann, On rings of operators. III, Ann. of Math. 41 (1940), 94-161. MR 898
  • 112. J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401-485. MR 29101
  • 113. A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, Paris, 1938.
  • 114. A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 247-302. MR 165033
  • 115. N. Wiener, Differential space, J. Math. Phys. M.I.T. 2 (1923), 131-174.
  • 116. Norbert Wiener, The Homogeneous Chaos, Amer. J. Math. 60 (1938), no. 4, 897–936. MR 1507356, https://doi.org/10.2307/2371268
  • 117. Norbert Wiener, The Fourier integral and certain of its applications, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1933 edition; With a foreword by Jean-Pierre Kahane. MR 983891


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DOI: https://doi.org/10.1090/S0002-9904-1965-11284-8

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