# zbMATH — the first resource for mathematics

A noncommutative probability theory. (English) Zbl 0407.46057

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 46K10 Representations of topological algebras with involution 47C10 Linear operators in $${}^*$$-algebras 81T05 Axiomatic quantum field theory; operator algebras 81S05 Commutation relations and statistics as related to quantum mechanics (general)
Full Text:
##### References:
 [1] Leo Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968. · Zbl 0174.48801 [2] C. D. Cushen and R. L. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probability 8 (1971), 454 – 469. · Zbl 0224.60049 [3] Alfons van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Functional Analysis 15 (1974), 378 – 393. · Zbl 0279.46027 [4] J. Dixmier, Les algèebres d’opérateurs dans l’espace Hilbertien (Algèebres de von Neumann), 2nd ed., Gauthier-Villars, Paris, 1969. · Zbl 0175.43801 [5] H. A. Dye, The Radon-Nikodým theorem for finite rings of operators, Trans. Amer. Math. Soc. 72 (1952), 243 – 280. · Zbl 0047.11101 [6] G. G. Emch, Algebraic methods in statistical mechanics and quantum field theory, Wiley, New York, 1972. · Zbl 0235.46085 [7] I. M. Gel$$^{\prime}$$fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. I. M. Gel$$^{\prime}$$fand and G. E. Shilov, Generalized functions. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions; Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. I. M. Gel$$^{\prime}$$fand and G. E. Shilov, Generalized functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967 [1977]. Theory of differential equations; Translated from the Russian by Meinhard E. Mayer. I. M. Gel$$^{\prime}$$fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. I. M. Gel$$^{\prime}$$fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. [8] S. Gudder and J.-P. Marchand, Noncommutative probability on von Neumann algebras, J. Mathematical Phys. 13 (1972), 799 – 806. · Zbl 0232.46064 · doi:10.1063/1.1666054 · doi.org [9] S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), no. 2, 427 – 436. · Zbl 0311.46015 [10] Paul R. Halmos and L. J. Savage, Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Ann. Math. Statistics 20 (1949), 225 – 241. · Zbl 0034.07502 [11] M. A. Naĭmark, Normed algebras, 3rd ed., Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the second Russian edition by Leo F. Boron; Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics. [12] M. Nakamura, M. Takesaki, and H. Umegaki, A remark on the expectations of operator algebras, Kōdai Math. Sem. Rep. 12 (1960), 82 – 90. · Zbl 0102.10802 [13] Robert T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85 – 124. · Zbl 0214.14102 [14] M. Reed and B. Simon, Methods of mathematical physics. Vol. 1 : Functional analysis, Academic Press, New York, 1972. · Zbl 0242.46001 [15] Robert Schatten, Norm ideals of completely continuous operators, Second printing. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 27, Springer-Verlag, Berlin-New York, 1970. · Zbl 0090.09402 [16] I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401 – 457. · Zbl 0051.34201 · doi:10.2307/1969729 · doi.org [17] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. · Zbl 0193.42502 [18] Hisaharu Umegaki, Conditional expectation in an operator algebra. II, Tôhoku Math. J. (2) 8 (1956), 86 – 100. · Zbl 0072.12501 · doi:10.2748/tmj/1178245011 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.