Communications in Mathematical Physics

, Volume 105, Issue 1, pp 123–131 | Cite as

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

  • Dénes Petz


A subalgebraM0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM0. The main result says thatM0 is weakly sufficient for (φ,ω) if and only ifM0 contains the Radon-Nikodym cocycle [Dφ,Dω] t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.


Entropy Neural Network Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Accardi, L., Cecchini, C.: Conditional expectation in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal.45, 245–273 (1982)Google Scholar
  2. 2.
    Araki, H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. Res. Inst. Math. Sci.9, 165–209 (1973)Google Scholar
  3. 3.
    Araki, H.: Recent developments in the theory of operator algebras and their significance in theoretical physics. Symposia Math., XX, 395–424. New York: Academic Press 1976Google Scholar
  4. 4.
    Araki, H.: Relative entropy of states of von Neumann algebras. I, II. Publ. Res. Inst. Math. Sci.11, 809–833 (1976) and13, 173–192 (1977)Google Scholar
  5. 5.
    Araki, H., Masuda, T.: Positive connes andL p-spaces for von Neumann algebras. Publ. Res. Inst. Math. Sci.18, 339–411 (1982)Google Scholar
  6. 6.
    Connes, A.: Une classification des facteurs de type III. Ann. Sci. Ec. Norm. Supér.6, 133–252 (1973)Google Scholar
  7. 7.
    Csiszár, I.: Information type measure of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung.2, 299–318 (1967)Google Scholar
  8. 8.
    Davis, C.: A Schwarz inequality for convex operator functions. Proc. Am. Math. Soc.8, 42–44 (1957)Google Scholar
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear operators. Part II. New York: Interscience 1958Google Scholar
  10. 10.
    Emch, G.G.: Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley 1972Google Scholar
  11. 11.
    Gudder, S., Merchand, J.P.: Noncommutative probability on von Neumann algebras. J. Math. Phys.13, 799–806 (1972)Google Scholar
  12. 12.
    Halmos, P.R., Savage, L.J.: Application of the Radon-Nikodym theorem to the theory of sufficient statistics. Ann. Math. Stat.20, 225–241 (1949)Google Scholar
  13. 13.
    Hiai, F., Ohya, M., Tsukada, M.: Sufficiency, KMS condition and relative entropy in von Neumann algebras. Pac. J. Math.96, 99–109 (1981)Google Scholar
  14. 14.
    Hiai, F., Ohya, M., Tsukada, M.: Sufficiency and relative entropy in *-algebras with applications in quantum systems. Pac. J. Math.107, 117–140 (1983)Google Scholar
  15. 15.
    Holevo, A.S.: Some estimates for the amount of information transmittable by a quantum communication channel (in Russian). Probl. Peredachi Inf.9, 3–11 (1973)Google Scholar
  16. 16.
    Holevo, A.S.: Investigations in the general theory of statistical decisions. Am. Math. Soc. Proc. Steklov Inst. Math.124 (1978)Google Scholar
  17. 17.
    Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  18. 18.
    Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat.22, 79–86 (1951)Google Scholar
  19. 19.
    Lieb, E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math.11, 267–288 (1973)Google Scholar
  20. 20.
    Lindblad, G.: Entropy, information and quantum measurement. Commun. Math. Phys.33, 305–322 (1973)Google Scholar
  21. 21.
    Lindblad, G.: Expectation and entropy inequalities for finite quantum systems. Commun. Math. Phys.39, 111–119 (1974)Google Scholar
  22. 22.
    Petz, D.: A dual in von Neumann algebras with weights. Q. J. Math. Oxf.35, 475–483 (1984)Google Scholar
  23. 23.
    Petz, D.: Properties of quantum entropy, quantum probability and applications II. Accardi, L., von Waldenfels, W. (eds.). Lecture Notes in Mathematics. Vol. 1136, pp. 428–441. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  24. 24.
    Petz, D.: Quasi-entropies for states of a von Neumann algebra. Publ. Res. Inst. Math. Sci.21, 787–800 (1985)Google Scholar
  25. 25.
    Raggio, G.A.: Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of a von Neumann algebra in standard form. Lett. Math. Phys.6, 233–236 (1982)Google Scholar
  26. 26.
    Strătilă, S., Zsidó, L.: Lectures on von Neumann algebras. Tunbridge Wells: Abacuss Press 1979Google Scholar
  27. 27.
    Strătilă, S.: Modular theory of operator algebras. Tunbridge Wells: Abacus Press 1981Google Scholar
  28. 28.
    Strasser, H.: Mathematical theory of statistics. Berlin: de Gruyter 1985Google Scholar
  29. 29.
    Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal.9, 306–321 (1972)Google Scholar
  30. 30.
    Umegaki, H.: Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep.14, 59–85 (1962)Google Scholar
  31. 31.
    Wolfe, J.C., Emch, G.G.:C*-algebraic formalism for coarse graining. I. General theory. J. Math. Phys.15, 1343–1347 (1974)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Dénes Petz
    • 1
  1. 1.Mathematisches Institut der Universität TübingenTübingenFederal Republic of Germany

Personalised recommendations