A Simple Singular Quantum Markov Semigroup

  • Franco Fagnola
Conference paper
Part of the Trends in Mathematics book series (TM)


We construct a quantum Markov semigroup on the von Neumann algebra B(L 2(ℝ+;ℂ)) of all bounded operators on a Hilbert space L 2(ℝ+;ℂ) with the following property: the biggest *-subalgebra of B(L 2(ℝ+;ℂ)) contained in the domain of the infinitesimal generator is not σ-weakly dense. Our semigroup is an extension of a classical Markov semigroup on L (ℝ+;ℂ) with the same property.


Manifold Kato 


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  1. [1]
    L. Accardi, A. Mohari: On the Structure of Classical and Quantum Flows. J. Funct. Anal. 135 (1996), 421–455.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    B.V.R. Bhat, F. Fagnola, K.B. Sinha: On quantum extensions of semi-groups of brownian motions on an half-line. Russian J. Math. Phys. 4 (1996), 13–28.MathSciNetMATHGoogle Scholar
  3. [3]
    O. Bratteli, D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics I. Springer-Verlag, New York, 1979.CrossRefGoogle Scholar
  4. [4]
    A.M. Chebotarev, F. Fagnola: Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups. J. Funct. Anal. 153 (1998), 382–404.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    F. Fagnola: Diffusion processes in Fock space. Quantum Probability and Related Topics IX (1994), 189–214.Google Scholar
  6. [6]
    F. Fagnola, R. Monte: A quantum extension of the semigroup Bessel processes. Mat. Zametki 60:5 (1996), 519–537.MathSciNetGoogle Scholar
  7. [7]
    F. Fagnola, R. Rebolledo: The approach to equilibrium of a class of quantum dynamical semigroups. Infinite Dimensional Analysis, Quantum Probability and Related Topics 1:4 (1998), 561–572.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    T. Kato: Perturbation theory for linear operators. Springer-Verlag, Berlin, 1966.MATHGoogle Scholar
  9. [9]
    A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1975.Google Scholar
  10. [10]
    W.F. Stinespring: Positive functions on C*-algebras, Proc. Am. Math. Soc., 6 (1955), 211–216.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Franco Fagnola
    • 1
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di GenovaGenovaItaly
  2. 2.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile

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