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Non-Archimedean Duality: Algebras, Groups, and Multipliers

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Abstract

We develop a duality theory for multiplier Banach-Hopf algebras over a non-Archimedean field 𝕂. As examples, we consider algebras corresponding to discrete groups and zero-dimensional locally compact groups with 𝕂-valued Haar measure, as well as algebras of operators generated by regular representations of discrete groups.

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References

  1. Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  2. Daws, M.: Multipliers, self-indiced and dual Banach algebras. Dissert. Math. 470, 62 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diarra, B.: La dualité de Tannaka dans les corps valués ultramétriques complets. C. R. Acad. Sci. Paris, Sér. A 279(26), 907–909 (1974)

    MathSciNet  MATH  Google Scholar 

  4. Diarra, B.: Sur quelques représentations p-adiques de \(\mathbb Z_{p}\). Indag. Math. 41, 481–493 (1979)

    MathSciNet  MATH  Google Scholar 

  5. Diarra, B.: Algèbres de Hopf et fonctions presque périodiques ultramétriques. Rivista di Mat. Pura ed Appl. 17, 113–132 (1996)

    MathSciNet  Google Scholar 

  6. Doran, R.S., Wichmann, J.: Approximate identities and factorization in Banach modules. Lect. Notes Math. 768, 305 (1979)

    MathSciNet  MATH  Google Scholar 

  7. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. 1. Springer, Berlin (1963)

    Book  MATH  Google Scholar 

  8. Johnson, B.E.: An introduction to the theory of centralizers. Proc. London Math. Soc. 14, 299–320 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  9. Katsaras, A.K., Beloyiannis, A.: Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J. 6(1), 33–44 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kisyński, J.: On Cohen’s proof of the factorization theorem. Ann. Polon. Math. 75(2), 177–192 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Kochubei, A.N.: On some classes of non-Archimedean operator algebras. Contemp. Math. 596, 133–148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kochubei, A.N.: Non-Archimedean group algebras with Baer reductions. Algebras Represent. Theory 17, 1861–1867 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. École Norm. Sup. 33, 837–934 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Kustermans, J., Vaes, S.: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92, 68–92 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Monna, A.F., Springer, T.A.: Intégration non-archimédienne. Indag. Math. 25, 634–653 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  16. Perez-Garcia, C., Schikhof, W.H.: Locally Convex Spaces over Non-Archimedean Valued Fields. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  17. van Rooij, A.C.M.: Non-Archimedean Functional Analysis. Marcel Dekker, New York (1978)

    MATH  Google Scholar 

  18. van Rooij, A.C.M., Schikhof, W. H.: Group representations in non-Archimedean Banach spaces. Bull. Soc. Math. France. Supplement Mémoire 39-40, 329–340 (1974)

    MathSciNet  MATH  Google Scholar 

  19. Schikhof, W.H.: Non-Archimedean Harmonic Analysis. Thesis, Nijmegen (1967)

    MATH  Google Scholar 

  20. Schikhof, W.H.: Non-Archimedean representations of compact groups. Compositio Math 23, 215–232 (1971)

    MathSciNet  MATH  Google Scholar 

  21. Serre, J.-P.: Endomorphismes complètement continus des espaces de Banach p-adiques. Publ. Math. IHES 12, 69–85 (1962)

    Article  Google Scholar 

  22. Soibelman, Y.: Quantum p-adic spaces and quantum p-adic groups. In: Geometry and Dynamics of Groups and Spaces, Progr. Math., vol. 265, pp. 697–719. Birkhäuser, Basel (2008)

  23. Sun, M.: Multiplier Hopf Algebras and Duality. Thesis. University of Windsor, Ontario, Canada. http://scholar.uwindsor.ca/etd/352 (2011)

  24. Timmermann, Th.: An Invitation to Quantum Groups and Duality. European Mathematical Society, Zürich (2008)

    Book  MATH  Google Scholar 

  25. Vaes, S., Van Daele, A.: Hopf C ∗-algebras. Proc. London Math. Soc. 82, 337–384 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Van Daele, A.: Multiplier Hopf algebras. Trans. Amer. Math. Soc. 342, 917–932 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Van Daele, A.: An algebraic framework for group duality. Adv. Math. 140, 323–366 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Van Daele, A.: Locally compact quantum groups. A von Neumann algebra approach. SIGMA 10 (2014). article 082

  29. Van Daele, A., Wang, S.: Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples. Banach Center Publ. 98, 367–415 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, J.-K.: Multipliers on commutative Banach algebras. Pacif. J. Math. 11, 1131–1149 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kiev, 01004, Ukraine

    Anatoly N. Kochubei

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  1. Anatoly N. Kochubei
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Correspondence to Anatoly N. Kochubei.

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Presented by Kenneth Goodearl.

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Kochubei, A.N. Non-Archimedean Duality: Algebras, Groups, and Multipliers. Algebr Represent Theor 19, 1081–1108 (2016). https://doi.org/10.1007/s10468-016-9612-9

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