Harmonic synthesis for subgroups

Carl S. Herz

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 3, page 91-123
  • ISSN: 0373-0956

Abstract

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Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p ( G ) , where A 2 ( G ) is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup H is amenable or normal. Global synthesis implies that each convolution operator on L p ( G ) with support in H which is the ultraweak limit of measures carried by H . The problem of passing from local to global synthesis is examined in an abstract context.

How to cite

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Herz, Carl S.. "Harmonic synthesis for subgroups." Annales de l'institut Fourier 23.3 (1973): 91-123. <http://eudml.org/doc/74143>.

@article{Herz1973,
abstract = {Let $G$ be a locally compact group and $H$ a closed subgroup. Then $H$ is always a set of local spectral synthesis with respect to the algebra $A_p(G)$, where $A_2(G)$ is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup $H$ is amenable or normal. Global synthesis implies that each convolution operator on $L^p(G)$ with support in $H$ which is the ultraweak limit of measures carried by $H$. The problem of passing from local to global synthesis is examined in an abstract context.},
author = {Herz, Carl S.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {91-123},
publisher = {Association des Annales de l'Institut Fourier},
title = {Harmonic synthesis for subgroups},
url = {http://eudml.org/doc/74143},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Herz, Carl S.
TI - Harmonic synthesis for subgroups
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 3
SP - 91
EP - 123
AB - Let $G$ be a locally compact group and $H$ a closed subgroup. Then $H$ is always a set of local spectral synthesis with respect to the algebra $A_p(G)$, where $A_2(G)$ is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup $H$ is amenable or normal. Global synthesis implies that each convolution operator on $L^p(G)$ with support in $H$ which is the ultraweak limit of measures carried by $H$. The problem of passing from local to global synthesis is examined in an abstract context.
LA - eng
UR - http://eudml.org/doc/74143
ER -

References

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  1. [1] P.J. COHEN, Factorization in group algebras, Duke Math. J. 26, (1959), 199-205. Zbl0085.10201MR21 #3729
  2. [2] J. DIXMIER, Les algèbres d'opérateurs dans l'espace Hilbertien, Gauthier-Villars, Paris, 2ème édition 1969. Zbl0175.43801
  3. [3] J. DIXMIER, Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris, 2ème édition 1969. Zbl0174.18601
  4. [4] C. DUNKL and D. RAMIREZ, Lp-multipliers supported by subgroups, Proc. Amer. Math. Soc. 34 (1972), 475-478. Zbl0253.43010
  5. [5] P. EYMARD, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. Zbl0169.46403MR37 #4208
  6. [6] P. EYMARD, Algèbres Ap et convoluteurs de Lp, Séminaire Bourbaki 367 (1969/1970). Zbl0264.43006
  7. [7] A. FIGA-TALAMANCA, Translation invariant operators in Lp, Duke Math. J. 32 (1965), 495-502. Zbl0142.10403MR31 #6095
  8. [8] C. HERZ, Remarques sur la Note précédente de M. Varopoulos, C.R. Acad. Sci. Paris A260 (1965), 6001-6004. Zbl0135.35404MR31 #6096
  9. [9] C. HERZ, Le rapport entre les algèbres Ap d'un groupe et d'un sous-groupe, C.R. Acad. Sci. Paris A271 (1970), 244-246. Zbl0195.13803MR42 #8307a
  10. [10] C. HERZ, Synthèse spectrale pour les sous-groupes par rapport aux algèbres Ap, C.R. Acad. Sci. Paris A271 (1970), 316-318. Zbl0195.13802MR42 #8307b
  11. [11] C. HERZ, The theory of p-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69-82. Zbl0216.15606MR42 #7833
  12. [12] G. MACKEY, Induced representations of locally compact groups, Annals of Math. 55 (1952), 101-139. Zbl0046.11601MR13,434a
  13. [13] H. MIRKIL, A counterexample to discrete spectral synthesis, Compositio Math. 14 (1960), 269-273. Zbl0099.10203MR23 #A4021
  14. [14] H. REITER, Classical Harmonic Analysis and Locally Compact Groups, Oxford 1968. Zbl0165.15601MR46 #5933
  15. [15] G.F. BACHELIS, W.A. PARKER, and K.A. Ross, Local units in L1(G), Proc. Amer. Math. Soc. 31 (1972), 312-313. Zbl0252.43013MR44 #5794

Citations in EuDML Documents

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  1. Ali Ghaffari, A generalization of amenability and inner amenability of groups
  2. Gilbert Arsac, Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire
  3. Gero Fendler, An L p -version of a theorem of D.A. Raikov
  4. Carl S. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes
  5. Christopher Meaney, Spherical functions and spectral synthesis
  6. Zhiguo Hu, Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups
  7. Edmond Granirer, On convolution operators with small support which are far from being convolution by a bounded measure
  8. Françoise Lust-Piquard, Means on C V p ( G ) -subspaces of C V p ( G ) with RNP and Schur property

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