On a class of convolution algebras of functions

Hans G. Feichtinger

Annales de l'institut Fourier (1977)

  • Volume: 27, Issue: 3, page 135-162
  • ISSN: 0373-0956

Abstract

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The Banach spaces Λ ( A , B , X , G ) defined in this paper consist essentially of those elements of L 1 ( G ) ( G being a locally compact group) which can in a certain sense be well approximated by functions with compact support. The main result of this paper is the fact that in many cases Λ ( A , B , X , G ) becomes a Banach convolution algebra. There exist many natural examples. Furthermore some theorems concerning inclusion results and the structure of these spaces are given. In particular we prove that simple conditions imply the existence of left approximate units or the density of K ( G ) . In the latter case a characterization of the closed left ideals is possible.

How to cite

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Feichtinger, Hans G.. "On a class of convolution algebras of functions." Annales de l'institut Fourier 27.3 (1977): 135-162. <http://eudml.org/doc/74324>.

@article{Feichtinger1977,
abstract = {The Banach spaces $\Lambda (A,B,X,G)$ defined in this paper consist essentially of those elements of $L^1(G)$ ($G$ being a locally compact group) which can in a certain sense be well approximated by functions with compact support. The main result of this paper is the fact that in many cases $\Lambda (A,B,X,G)$ becomes a Banach convolution algebra. There exist many natural examples. Furthermore some theorems concerning inclusion results and the structure of these spaces are given. In particular we prove that simple conditions imply the existence of left approximate units or the density of $K(G)$. In the latter case a characterization of the closed left ideals is possible.},
author = {Feichtinger, Hans G.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {135-162},
publisher = {Association des Annales de l'Institut Fourier},
title = {On a class of convolution algebras of functions},
url = {http://eudml.org/doc/74324},
volume = {27},
year = {1977},
}

TY - JOUR
AU - Feichtinger, Hans G.
TI - On a class of convolution algebras of functions
JO - Annales de l'institut Fourier
PY - 1977
PB - Association des Annales de l'Institut Fourier
VL - 27
IS - 3
SP - 135
EP - 162
AB - The Banach spaces $\Lambda (A,B,X,G)$ defined in this paper consist essentially of those elements of $L^1(G)$ ($G$ being a locally compact group) which can in a certain sense be well approximated by functions with compact support. The main result of this paper is the fact that in many cases $\Lambda (A,B,X,G)$ becomes a Banach convolution algebra. There exist many natural examples. Furthermore some theorems concerning inclusion results and the structure of these spaces are given. In particular we prove that simple conditions imply the existence of left approximate units or the density of $K(G)$. In the latter case a characterization of the closed left ideals is possible.
LA - eng
UR - http://eudml.org/doc/74324
ER -

References

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  1. [1] B.A. BARNES, Banach algebras which are ideals in a Banach algebra, Pac. J. Math., 38 (1971), 1-7. Zbl0226.46054MR46 #9738
  2. [2] P.L. BUTZER-K. SCHERER, Approximationsprozesse und Interpolationsmethoden, Bibl. Inst. Mannheim, 1968. Zbl0177.08501
  3. [3] Y. DOMAR, Harmonic analysis based on certain commutative Banach algebras, Acta Math., 96 (1956), 1-66. Zbl0071.11302MR17,1228a
  4. [4] H.G. FEICHTINGER, Some new subalgebras of L1 (G), Indag. Math., 36 (1974), 44-47. Zbl0272.43004MR49 #1006
  5. [5] E. HILLE, Functional analysis and semigroups, Amer. Math. Soc. Publ., XXXI (1948). Zbl0033.06501MR9,594b
  6. [6] A. HULANICKI, On the spectrum of convolution operators on groups of polynomial growth, Invent. math., 17 (1972), 135-142. Zbl0264.43007MR48 #2304
  7. [7] H. REITER, Classical harmonic analysis and locally compact groups, Oxford University Press, 1968. Zbl0165.15601MR46 #5933
  8. [8] R. SPECTOR, Sur la structure locale des groupes abéliens localement compacts, Bull. Soc. Math. France, Mémoire 24 (1970). Zbl0215.18603MR44 #729
  9. [9] H. CH. WANG, Nonfactorization in group algebras, Studia math., 42 (1972), 231-241. Zbl0273.43008MR46 #2355
  10. [10] L.H. BRANDENBURG, On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl., 50 (1975), 489-510. Zbl0302.46042MR51 #13695
  11. [11] I.I. HIRSCHMANN, Finite sections of Wiener-Hopf equations and Szegö polynomials, J. Math. Anal. Appl., 11 (1965), 290-320. Zbl0173.42601MR31 #6133

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