Abstract
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.
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Perron, O.: Grundlagen für eine Theorie der Jacobischen Kettenbruchalgorithmen. Math. Ann.64, 1–76 (1907)
Perron, O.: Zur Theorie der Matrices. Math. Ann.64, 248–263 (1908)
Frobenius, F. G.: Über Matrizen aus positiven Elementen, I, II. Sitzungsber. Akad. Wiss. Berlin, Phys. Math. kl. 471–476 (1908); 514–518 (1909)
Frobenius, F. G.: Über Matrizen aus nicht negativen Elementen. Sitzungsber. Akad. Wiss. Berlin, Phys. Math. kl. 456–477 (1912)
Gantmacher, F. R.: Applications of the theory of matrices. New York: Interscience 1959
Evans, D., Høegh-Krohn, R.: Spectral properties of positive maps on C*-algebras. J. London Math. Soc.17, 345–355 (1978)
Krein, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Am. Math. Soc. Transl.10, 199–235 (1950)
Jentzsch, R.: Über Integralgleichungen mit positivem Kern. J. reine angew. Math.141, 235–244 (1912)
Karlin, S.: Positive operators. J. Math. Mech.8, 907–937 (1959)
Rota, G. C.: On the eigenvalues of positive operators. Bull. Am. Math. Soc.67, 556–558 (1961)
Schaefer, H. H.: Banach lattices and positive operators. Berlin, Heidelberg, New York: Springer 1974
Koopman, B. O.: Hamiltonian systems and transformations in Hilbert spaces. Proc. Nat. Acad. Sci.17, 315–318 (1931)
Carleman, T.: Application de la théorie des équations intégrales singulières aux équations différentielles de la dynamique. Ark. Mat., Astr. Fys.22B, No. 7 (1931)
von Neumann, J.: Zur Operatorenmethode in der klassischen Mechanik. Ann. Math.33, 587–642 (1932)
Jacobs, K.: Lecture notes on ergodic theory. Aarhus University (1962/63)
Størmer, E.: Spectra of ergodic transformations. J. Funct. Anal.15, 202–215 (1974)
Størmer, E.: Spectral subspaces of automorphisms. In: Rendic. S. I. F., Varenna, LX, D. Kastler (ed.), pp. 128–138. New York: Academic Press 1976
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 1976
Kastler, D.: Equilibrium states of matter and operator algebras. In: Symposia mathematica XX, Ist. Naz. Alta Mat., pp. 49–107. London: Academic Press 1976
Evans, D., Sund, T.: Spectral subspaces for compact actions. Preprint, Oslo University (1977)
Kadison, R.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math.56, 494–503 (1952)
Choi, M. D.: A Schwarz inequality for positive linear maps on C*-algebras. III. J. Math.18, 565–574 (1974)
Størmer, E.: Positive linear maps on operator algebras. Acta Math.110, 233–278 (1963)
Størmer, E.: Positive linear maps of C*-algebras. In: Lecture notes in physics, Vol. 29, pp. 85–106. Berlin, Heidelberg, New York: Springer 1974
Evans, D. E.: Irreducible quantum dynamical semigroups. Preprint, Oslo University (1976)
Davies, E. D.: Quantum theory for open systems. New York: Academic Press 1976
Emch, G. G.: Non abelian special K-flows. J. Funct. Anal.19, 1–12 (1975)
Emch, G. G.: Generalized K-flows. Commun. math. Phys.49, 191–215 (1976)
Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and Markov semigroups on C*-algebras. Commun. math. Phys.56, 173–187 (1977)
Gorini, V., Frigerio, A., Verri, M., Kossakowski, A., Sudarshan, E. C. G.: Properties of markovian master equations. Preprint, University of Texas (1977)
Emch, G. G., Albeverio, S., Eckman, J. P.: Quasi free generalized K-flows. Rept. Math. Phys.13, 73–85 (1978)
Albeverio, S., Høegh-Krohn, R.: Ergodic actions of compact groups on von Neumann algebras. Preprint (Nov. 1977)
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Albeverio, S., Høegh-Krohn, R. Frobenius theory for positive maps of von Neumann algebras. Commun.Math. Phys. 64, 83–94 (1978). https://doi.org/10.1007/BF01940763
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DOI: https://doi.org/10.1007/BF01940763