Revista Matemática Complutense

, Volume 29, Issue 1, pp 13–57 | Cite as

The \(C^{*}\)-algebras of connected real two-step nilpotent Lie groups

  • Janne-Kathrin GüntherEmail author
  • Jean Ludwig


Using the operator valued Fourier transform, the \(C^{*}\)-algebras of connected real two-step nilpotent Lie groups are characterized as algebras of operator fields defined over their spectra. In particular, it is shown by explicit computations, that the Fourier transform of such \(C^{*}\)-algebras fulfills the norm controlled dual limit property.


\(C^{*}\)-algebra Two-step nilpotent Lie group Fourier transform Norm controlled dual limit property 

Mathematics Subject Classification

43 (Abstract Harmonic Analysis) 



This work is supported by the Fonds National de la Recherche, Luxembourg (Project Code 3964572).


  1. 1.
    Brown, I.: Dual topology of a nilpotent Lie group. Ann. Sci. l’É.N.S. 4e Sér. Tome 6(3), 407–411 (1973)Google Scholar
  2. 2.
    Corwin, L., Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. In: Cambridge Studies in Advanced Mathematics, vol. 18. Cambridge University Press, Cambridge (1990)Google Scholar
  3. 3.
    Dixmier, J.: \(C^{*}\)-algebras. Translated from French by Francis Jellett. In: North-Holland Mathematical Library, vol. 15. North-Holland, Amsterdam (1977)Google Scholar
  4. 4.
    Lahiani, R.: Analyse Harmonique sur certains groupes de Lie à croissance polynomiale. Ph.D. thesis, University of Luxembourg and Université Paul Verlaine-Metz (2010)Google Scholar
  5. 5.
    Leptin, H., Ludwig, J.: Unitary representation theory of exponential Lie groups. In: De Gruyter Expositions in Mathematics, vol. 18. Berlin (1994)Google Scholar
  6. 6.
    Lin, Y.-F., Ludwig, J.: The \(C^{*}\)-algebras of \(ax+b\)-like groups. J. Funct. Anal. 259, 104–130 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ludwig, J., Turowska, L.: The \(C^{*}\)-algebras of the Heisenberg Group and of thread-like Lie groups. Math. Z. 268(3–4), 897–930 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ludwig, J., Zahir, H.: On the nilpotent \(*\)-Fourier transform. Lett. Math. Phys. 30, 23–24 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pukanszky, L.: Leçons sur les représentations des groupes. Dunod, Paris (1967)zbMATHGoogle Scholar
  10. 10.
    Regeiba, H.: Les \(C^{*}\). Ph.D. thesis, Université de Lorraine (2014)Google Scholar
  11. 11.
    Regeiba, H., Ludwig, J.: \(C^{*}\)-Algebras with Norm Controlled Dual Limits and Nilpotent Lie Groups (2013). arXiv:1309.6941

Copyright information

© Universidad Complutense de Madrid 2015

Authors and Affiliations

  1. 1.Mathematical Research UnitUniversité du LuxembourgLuxembourgLuxembourg
  2. 2.UMR 7502, Institut Elie Cartan de LorraineUniversité de LorraineMetz Cedex 01France

Personalised recommendations