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The primitive spectrum of a semigroup of Markov operators

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Abstract

For a semigroup \(\mathcal {S}\) of Markov operators on a space of continuous functions, we use \(\mathcal {S}\)-invariant ideals to describe qualitative properties of \(\mathcal {S}\) such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive\(\mathcal {S}\)-ideals and endow the space of those ideals with an appropriate topology. This approach is inspired by the representation theory of C*-algebras and can be adapted to our dynamical setting. In the particularly important case of Koopman semigroups, we characterize the centers of attraction of the underlying dynamical system in terms of the invariant ideal structure of \(\mathcal {S}\).

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Notes

  1. Remark 3.8 and Example 3.9 (i) were kindly suggested by the referee.

References

  1. Berglund, J.F., Junghenn, H., Milnes, P.: Analysis on Semigroups. Function Spaces, Compactifications, Representations. Wiley, Hoboken (1989)

    MATH  Google Scholar 

  2. Bernard, R.R.: Probability in dynamical transformation groups. Duke Math. J. 18, 307–319 (1951)

    Article  MathSciNet  Google Scholar 

  3. Blackadar, B.: Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Springer, Berlin (2006)

    MATH  Google Scholar 

  4. Chen, Z., Dai, X.: Chaotic dynamics of minimal center of attraction for a flow with discrete amenable phase group. J. Math. Anal. Appl. 456, 1397–1414 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chou, C.: Minimal sets and ergodic measures for \(\beta N{\setminus } {N}\). Illinois J. Math. 13, 777–788 (1967)

    Article  MathSciNet  Google Scholar 

  6. Dai, X.: On chaotic minimal center of attraction of a Lagrange stable motion for topological semi flows. J. Differ. Equ. 260, 4393–4409 (2016)

    Article  MathSciNet  Google Scholar 

  7. Day, M.M.: Fixed-point theorems for compact convex sets. Illinois J. Math. 5, 585–590 (1961)

    Article  MathSciNet  Google Scholar 

  8. Dixmier, J.: C*-Algebras. North Holland, Amsterdam (1977)

    MATH  Google Scholar 

  9. Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Springer, London (2011)

    MATH  Google Scholar 

  10. Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory. Springer, Cham (2015)

    Book  Google Scholar 

  11. Görtz, U., Wedhorn, T.: Algebraic Geometry I. Vieweg+Teubner, Berlin (2010)

    Book  Google Scholar 

  12. Hilmy, H.: Sur les centres d’attraction minimaux des systèmes dynamiques. Compos. Math. 3, 227–238 (1936)

    MathSciNet  MATH  Google Scholar 

  13. Hofmann, K.H.: The Dauns–Hofmann theorem revisited. J. Algebra Appl. 10, 29–37 (2011)

    Article  MathSciNet  Google Scholar 

  14. Jacobs, K., Rosenmüller, J.: Selecta Mathematica IV. Springer, Cham (1972)

    Book  Google Scholar 

  15. Kelley, J.L.: General Topology. Springer, New York (1975)

    MATH  Google Scholar 

  16. Mané, R.: Ergodic Theory and Differentiable Dynamics. Springer, Berlin (1987)

    Book  Google Scholar 

  17. Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)

    Book  Google Scholar 

  18. Nagel, R.: Mittelergodische Halbgruppen linearer Operatoren. Ann. Inst. Fourier 23, 75–87 (1973)

    Article  MathSciNet  Google Scholar 

  19. Parry, W.: Topics in Ergodic Theory. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  20. Paterson, A.L.T.: Amenability. American Mathematical Society, Providence (1988)

    Book  Google Scholar 

  21. Pedersen, G.K.: C*-Algebras and Their Automorphism Groups. Academic Press, Cambridge (1979)

    MATH  Google Scholar 

  22. Phelps, R.R.: Lectures on Choquet’s Theorem, 2nd edn. Springer, Berlin (2001)

    Book  Google Scholar 

  23. Rosenblatt, M.: Transition probability operators. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability. Part 2, vol. 2, pp. 473–483. University of California Press (1976)

  24. Schaefer, H.H.: Invariant ideals of positive operators in \({\rm C}(X)\). I. Illinois J. Math. 11, 703–715 (1967)

    Article  MathSciNet  Google Scholar 

  25. Schaefer, H.H.: Invariant ideals of positive operators in \({\rm C}(X)\). II. Illinois J. Math. 12, 525–538 (1968)

    Article  MathSciNet  Google Scholar 

  26. Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)

    Book  Google Scholar 

  27. Schreiber, M.: Uniform families of ergodic operator nets. Semigroup Forum 86, 321–336 (2013)

    Article  MathSciNet  Google Scholar 

  28. Sigmund, K.: On minimal centers of attraction and generic points. J. Reine Angew. Math. 295, 72–79 (1977)

    MathSciNet  MATH  Google Scholar 

  29. Sine, R.: Geometric theory of a single Markov operator. Pac. J. Math. 27, 155–166 (1968)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author wants to thank Roland Derndinger, Nikolai Edeko, Ulrich Groh and Rainer Nagel for ideas, suggestions and inspiring discussions. The author is also very grateful to the referee for their suggestions. Their thorough reports have led to major improvements of the article. In particular, their valuable advice vastly helped to improve the abstract, the introduction and the readability of the article overall.

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Correspondence to Henrik Kreidler.

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Kreidler, H. The primitive spectrum of a semigroup of Markov operators. Positivity 24, 287–312 (2020). https://doi.org/10.1007/s11117-019-00678-0

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