Communications in Mathematical Physics

, Volume 64, Issue 1, pp 83–94 | Cite as

Frobenius theory for positive maps of von Neumann algebras

  • Sergio Albeverio
  • Raphael Høegh-Krohn


Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. 1. a)
    Perron, O.: Grundlagen für eine Theorie der Jacobischen Kettenbruchalgorithmen. Math. Ann.64, 1–76 (1907)Google Scholar
  2. 1. b)
    Perron, O.: Zur Theorie der Matrices. Math. Ann.64, 248–263 (1908)Google Scholar
  3. 2. a)
    Frobenius, F. G.: Über Matrizen aus positiven Elementen, I, II. Sitzungsber. Akad. Wiss. Berlin, Phys. Math. kl. 471–476 (1908); 514–518 (1909)Google Scholar
  4. 2. b)
    Frobenius, F. G.: Über Matrizen aus nicht negativen Elementen. Sitzungsber. Akad. Wiss. Berlin, Phys. Math. kl. 456–477 (1912)Google Scholar
  5. 3.
    Gantmacher, F. R.: Applications of the theory of matrices. New York: Interscience 1959Google Scholar
  6. 4.
    Evans, D., Høegh-Krohn, R.: Spectral properties of positive maps on C*-algebras. J. London Math. Soc.17, 345–355 (1978)Google Scholar
  7. 5.
    Krein, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Am. Math. Soc. Transl.10, 199–235 (1950)Google Scholar
  8. 6.
    Jentzsch, R.: Über Integralgleichungen mit positivem Kern. J. reine angew. Math.141, 235–244 (1912)Google Scholar
  9. 7.
    Karlin, S.: Positive operators. J. Math. Mech.8, 907–937 (1959)Google Scholar
  10. 8.
    Rota, G. C.: On the eigenvalues of positive operators. Bull. Am. Math. Soc.67, 556–558 (1961)Google Scholar
  11. 9.
    Schaefer, H. H.: Banach lattices and positive operators. Berlin, Heidelberg, New York: Springer 1974Google Scholar
  12. 10.
    Koopman, B. O.: Hamiltonian systems and transformations in Hilbert spaces. Proc. Nat. Acad. Sci.17, 315–318 (1931)Google Scholar
  13. 11.
    Carleman, T.: Application de la théorie des équations intégrales singulières aux équations différentielles de la dynamique. Ark. Mat., Astr. Fys.22B, No. 7 (1931)Google Scholar
  14. 12.
    von Neumann, J.: Zur Operatorenmethode in der klassischen Mechanik. Ann. Math.33, 587–642 (1932)Google Scholar
  15. 13.
    Jacobs, K.: Lecture notes on ergodic theory. Aarhus University (1962/63)Google Scholar
  16. 14. a)
    Størmer, E.: Spectra of ergodic transformations. J. Funct. Anal.15, 202–215 (1974)Google Scholar
  17. 14. b)
    Størmer, E.: Spectral subspaces of automorphisms. In: Rendic. S. I. F., Varenna, LX, D. Kastler (ed.), pp. 128–138. New York: Academic Press 1976Google Scholar
  18. 15.
    Olesen, D.: On spectral subspaces and their applications to automorphism groups. In: Symposia mathematica XX, Ist. Naz. Alta. Mat., pp. 253–296. London: Academic Press 1976Google Scholar
  19. 16.
    Kastler, D.: Equilibrium states of matter and operator algebras. In: Symposia mathematica XX, Ist. Naz. Alta Mat., pp. 49–107. London: Academic Press 1976Google Scholar
  20. 17.
    Evans, D., Sund, T.: Spectral subspaces for compact actions. Preprint, Oslo University (1977)Google Scholar
  21. 18.
    Kadison, R.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math.56, 494–503 (1952)Google Scholar
  22. 19.
    Choi, M. D.: A Schwarz inequality for positive linear maps on C*-algebras. III. J. Math.18, 565–574 (1974)Google Scholar
  23. 20. a)
    Størmer, E.: Positive linear maps on operator algebras. Acta Math.110, 233–278 (1963)Google Scholar
  24. 20. b)
    Størmer, E.: Positive linear maps of C*-algebras. In: Lecture notes in physics, Vol. 29, pp. 85–106. Berlin, Heidelberg, New York: Springer 1974Google Scholar
  25. 21.
    Evans, D. E.: Irreducible quantum dynamical semigroups. Preprint, Oslo University (1976)Google Scholar
  26. 22.
    Davies, E. D.: Quantum theory for open systems. New York: Academic Press 1976Google Scholar
  27. 23. a)
    Emch, G. G.: Non abelian special K-flows. J. Funct. Anal.19, 1–12 (1975)Google Scholar
  28. 23. b)
    Emch, G. G.: Generalized K-flows. Commun. math. Phys.49, 191–215 (1976)Google Scholar
  29. 24.
    Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and Markov semigroups on C*-algebras. Commun. math. Phys.56, 173–187 (1977)Google Scholar
  30. 25.
    Gorini, V., Frigerio, A., Verri, M., Kossakowski, A., Sudarshan, E. C. G.: Properties of markovian master equations. Preprint, University of Texas (1977)Google Scholar
  31. 26.
    Emch, G. G., Albeverio, S., Eckman, J. P.: Quasi free generalized K-flows. Rept. Math. Phys.13, 73–85 (1978)Google Scholar
  32. 27.
    Albeverio, S., Høegh-Krohn, R.: Ergodic actions of compact groups on von Neumann algebras. Preprint (Nov. 1977)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Sergio Albeverio
    • 1
  • Raphael Høegh-Krohn
    • 1
  1. 1.Institute of MathematicsUniversity of OsloBlindern, OsloNorway

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