Abstract
The author introduces the concepts of positive-definite and conditionally positive definite functions with values in the algebra of bounded maps of a C*-algebra. An analog of Schoenberg's theorem is proved, a GNS-representation is obtained for conditionally positive-definite functions in terms of suitable cocycles, and this representation leads to a noncommutative generalization of the Lévy— Khinchin formula. Applications to the problem of continuous measurement in quantum mechanics are considered. A complete mathematical description is presented of continuous measurement processes, based on the analogy with the classical parts of probability theory—the theory of infinitely divisible distributions and functional limit theorems for processes with independent increments.
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Literature cited
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968) [Russian translation], Nauka, Moscow (1977)].
I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions [in Russian], Vol. 4, Nauka, Moscow (1961).
B. V. Gnedenko and A. N. Kolmogorov, Limit Theorems for Sums of Independent Random Variables [in Russian], Gostekhizdat, Moscow-Leningrad (1949).
M. B. Menskii, The Path Group: Measurements, Fields, Particles [in Russian], Nauka, Moscow (1983).
J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin (1932).
H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin (1977).
A. S. Kholevo, Probability and Statistical Aspects of Quantum Theory [in Russian], Nauka, Moscow (1980).
A. S. Kholevo, “On measurements of parameters of a quantum stochastic process,” Teor. Mat. Fiz.,57, No. 3, 424–437 (1983).
A. S. Kholevo, “Infinitely divisible measurements in quantum probability theory,” Teor. Veroyatn. Primen.,31, No. 3, 560–564 (1986).
A. S. Kholevo, “Representations of Lévy—Khinchin type in quantum probability theory,” Teor. Veroyatn. Primen.,32, No. 1, 142–146 (1987).
A. Barchielli, L. Lanz, and G. M. Prosperi, “A model of macroscopic description and continual observations in quantum mechanics,” Nuovo Cimento,72B, 79–121 (1982).
A. Barchielli, L. Lanz, and G. M. Prosperi, “Statistics of continuous trajectories in quantum mechanics: operation-valued stochastic processes,” Found. Phys.,13, 779–812 (1983).
A. Barchielli and G. Lupieri, “Dilations of operation-valued stochastic processes,” Lect. Notes Math.,1136, 57–66 (1985).
A. Barchielli, “Continuous observations in quantum mechanics: an application to gravitational-wave detectors,” Phys. Rev. D,32, 347–367 (1985).
E. Christensen and D. E. Evans, “Cohomology of operator algebras and quantum dynamical semigroups,” J. London Math. Soc.,20, No. 2, 358–368 (1979).
E. B. Davies, Quantum Theory of Open Systems, Academic Press, London (1976).
J. Dixmier, Les Algèbres d'Opérateurs Dans L'Espace Hilbertien, Gauthier-Villars, Paris (1969).
D. E. Evans and J. T. Lewis, Dilations of Irreversible Evolutions in Algebraic Quantum Theory (Commun. Dublin Inst. Advanced Studies, Ser. A,24) (1977).
A. Guichardet, Symmetric Hilbert Spaces and Related Topics Springer, Berlin (1972).
A. S. Holevo, “Conditionally positive definite functions in quantum probability,” in: Proc. Int. Congress Math., Berkeley, California, U.S.A., 1986 (1987), pp. 1011–1020.
S. Johansen, “An application of extreme point method to the representation of infinitely divisible distributions,” Z. Wahrscheinlichkeitst. Geb.,5, 304–316 (1966).
G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys.,48, No. 2, 119–130 (1976).
B. Misra and E. C. G. Sudarshan, “The Zeno's paradox in quantum theory,” J. Math. Phys.,18, No. 4, 756–763 (1977).
M. Ozawa, “Quantum measuring processes of continuous observables,” J. Math. Phys.,25, No. 1, 79–87 (1984).
K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems, Springer, Berlin (1972).
K. R. Parthasarathy, “One parameter semigropus of completely positive maps on groups arising from quantum stochastic differential equations,” Bull. Unione Matem. Italiana,5A, 57–66 (1986).
Quantum Probability and Applications II, Springer, Berlin (1985).
D. Russo and H. A. Dye, “A note on unitary operators in C*-algebras,” Duke Math. J.,33, 413–416 (1966).
I. J. Schoenberg, “Metric spaces and positive definite functions,” Trans. Am. Math. Soc.,4, 522–530 (1938).
M. Schürmann, “Positive and conditionally positive linear functionals,” in: Quantum Probability and Applications II, Springer, Berlin (1985), pp. 474–492. pp. 474–492.
M. Takesaki, Theory of Operator Algebras, Springer, New York (1979).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 103–147, 1990.
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Kholevo, A.S. Conditionally positive-definite functions in quantum probability theory. J Math Sci 56, 2670–2697 (1991). https://doi.org/10.1007/BF01095976
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DOI: https://doi.org/10.1007/BF01095976