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. Carl S. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes
  4. Gero Fendler, An L p -version of a theorem of D.A. Raikov
  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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