Hull-minimal ideals in the Schwartz algebra of the Heisenberg group

J. Ludwig

Studia Mathematica (1998)

  • Volume: 130, Issue: 1, page 77-98
  • ISSN: 0039-3223

Abstract

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For every closed subset C in the dual space Ĥ n of the Heisenberg group H n we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra S ( H n ) and we show that in general for two closed subsets C 1 , C 2 of Ĥ n the product of j ( C 1 ) and j ( C 2 ) is different from j ( C 1 ∩ C 2 ) .

How to cite

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Ludwig, J.. "Hull-minimal ideals in the Schwartz algebra of the Heisenberg group." Studia Mathematica 130.1 (1998): 77-98. <http://eudml.org/doc/216542>.

@article{Ludwig1998,
abstract = {For every closed subset C in the dual space $Ĥ_n$ of the Heisenberg group $H_n$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S(H_n)$ and we show that in general for two closed subsets $C_1, C_2$ of $Ĥ_n$ the product of $j(C_1)$ and $j(C_2)$ is different from $j(C_1∩C_2)$.},
author = {Ludwig, J.},
journal = {Studia Mathematica},
keywords = {Fréchet algebra; dual space; Heisenberg group; Fourier transform; hull-minimal ideal; Schwartz algebra},
language = {eng},
number = {1},
pages = {77-98},
title = {Hull-minimal ideals in the Schwartz algebra of the Heisenberg group},
url = {http://eudml.org/doc/216542},
volume = {130},
year = {1998},
}

TY - JOUR
AU - Ludwig, J.
TI - Hull-minimal ideals in the Schwartz algebra of the Heisenberg group
JO - Studia Mathematica
PY - 1998
VL - 130
IS - 1
SP - 77
EP - 98
AB - For every closed subset C in the dual space $Ĥ_n$ of the Heisenberg group $H_n$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S(H_n)$ and we show that in general for two closed subsets $C_1, C_2$ of $Ĥ_n$ the product of $j(C_1)$ and $j(C_2)$ is different from $j(C_1∩C_2)$.
LA - eng
KW - Fréchet algebra; dual space; Heisenberg group; Fourier transform; hull-minimal ideal; Schwartz algebra
UR - http://eudml.org/doc/216542
ER -

References

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  1. [BD] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. Zbl0271.46039
  2. [CG] L. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications Cambridge Univ. Press, 1990. 
  3. [Di1] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969. 
  4. [Di2] J. Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Etudes Sci. Publ. Math. 6 (1960), 305-317. Zbl0100.32303
  5. [FO] G. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989. 
  6. [FS] G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton Univ. Press, 1982. Zbl0508.42025
  7. [Hu] A. Hulanicki, A functional calculus for Rockland oerators on nilpotent Lie groups, Studia Math. 78 (1984), 253-266. Zbl0595.43007
  8. [Lu1] J. Ludwig, Polynomial growth and ideals in group algebras, Manuscripta Math. 30 (1980), 215-221. Zbl0417.43005
  9. [Lu2] J. Ludwig, Minimal C*-dense ideals and algebraically irreducible representations of the Schwartz-algebra of a nilpotent Lie group, in: Harmonic Analysis, Springer, 1987, 209-217. 
  10. [Lu3] J. Ludwig, Topologically irreducible of the Schwartz-algebra a nilpotent Lie group, Arch. Math. (Basel) 54 (1990), 284-292. Zbl0664.43002
  11. [LM] J. Ludwig et C. Molitor-Braun, L'algèbre de Schwartz d'un groupe de Lie nilpotent, Travaux de Mathématiques Publications du C.U. Luxembourg, 1996. 
  12. [LRS] J. Ludwig, G. Rosenbaum and J. Samuel, The elements of bounded trace in the C*-algebra of a nilpotent Lie group, Invent. Math. 83 (1986), 167-190. Zbl0587.22003
  13. [LZ] J. Ludwig and H. Zahir, On the nilpotent Fourier transform, Lett. math. Phys. 30, (1994), 23-34. Zbl0798.22004
  14. [Rei] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Math. Monographs, Oxford Univ. Press, 1968. Zbl0165.15601

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