Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A noncommutative probability theory


Authors: S. P. Gudder and R. L. Hudson
Journal: Trans. Amer. Math. Soc. 245 (1978), 1-41
MSC: Primary 46L55; Secondary 46K99, 81B99
DOI: https://doi.org/10.1090/S0002-9947-1978-0511398-0
MathSciNet review: 511398
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A noncommutative probability theory is developed in which no boundedness, finiteness, or ``tracial'' conditions are imposed. The underlying structure of the theory is a ``probability algebra'' $ (\mathcal{a},\omega )$ where $ \mathcal{a}$ is a *-algebra and $ \omega $ is a faithful state on $ \mathcal{a}$. Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.


References [Enhancements On Off] (What's this?)

  • [1] L. Breiman, Probability, Addison-Wesley, Reading, Mass., 1968. MR 0229267 (37:4841)
  • [2] C. D. Cushen and R. L. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probability 8 (1971), 454-469. MR 0289082 (44:6277)
  • [3] A. van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Functional Analysis 15 (1974), 378-393. MR 0346539 (49:11264)
  • [4] J. Dixmier, Les algèebres d'opérateurs dans l'espace Hilbertien (Algèebres de von Neumann), 2nd ed., Gauthier-Villars, Paris, 1969.
  • [5] H. A. Dye, The Radon-Nikodým theorem for finite rings of operators, Trans. Amer. Math. Soc. 72 (1952), 243-280. MR 0045954 (13:662b)
  • [6] G. G. Emch, Algebraic methods in statistical mechanics and quantum field theory, Wiley, New York, 1972.
  • [7] I. M. Gel'fand and N. Ya. Vilenkin, Generalized functions, Vol. 4, Academic Press, New York, 1964. MR 0435834 (55:8786d)
  • [8] S. P. Gudder and J.-P. Marchand, Probability theory on von Neumann algebras, J. Mathematical Phys. 13 (1972), 799-806. MR 0302108 (46:1261)
  • [9] S. P. Gudder and D. Strawther, Orthogonally additive and orthogonally monotone functions on vector spaces, Pacific J. Math. 58 (1975), 427-436. MR 0390719 (52:11542)
  • [10] P. R. Halmos and L. J. Savage, Application of the Radon-Nikodým theorem to the theory of sufficient statistics, Ann. Math. Statist. 20 (1949), 225-241. MR 0030730 (11:42g)
  • [11] M. A. Naimark, Normed algebras, 3rd ed., Wolters-Noordhoff, Gröningen, 1972. MR 0438123 (55:11042)
  • [12] M. Nakumara, M. Takesaki and H. Umegaki, A remark on the expectations of operator algebras, Kōdai Math. Sem. Rep. 12 (1960), 82-90. MR 0250084 (40:3325)
  • [13] R. T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85-124. MR 0283580 (44:811)
  • [14] M. Reed and B. Simon, Methods of mathematical physics. Vol. 1 : Functional analysis, Academic Press, New York, 1972.
  • [15] R. Schatten, Norm ideals of completely continuous operators, Springer, Berlin, 1970. MR 0257800 (41:2449)
  • [16] I. E. Segal, Non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-457. MR 0054864 (14:991f)
  • [17] M. Takesaki, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math., vol. 128, Springer-Verlag, Berlin, 1970. MR 0270168 (42:5061)
  • [18] H. Umegaki, Conditional expectation in an operator algebra. II, Tôhoku Math. J. 8 (1956), 86-100. MR 0090789 (19:872b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L55, 46K99, 81B99

Retrieve articles in all journals with MSC: 46L55, 46K99, 81B99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0511398-0
Keywords: Noncommutative probability, quantum probability, quantum mechanics, *-algebras, unbounded representations, commutants
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society