On a class of convolution algebras of functions
Annales de l'Institut Fourier, Volume 27 (1977) no. 3, pp. 135-162.

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.

Les espaces de Banach Λ(A,B,X,G) définis dans cette note sont des sous-espaces de L 1 (G) (G étant un groupe localement compact). Ils contiennent essentiellement des éléments de L 1 (G) qui peuvent dans un certain sens être bien approchés par des fonctions à support compact. Le résultat principal est le fait que Λ(A,B,X,G) est souvent une algèbre de Banach avec convolution comme multiplication. Il existe un grand nombre d’exemples très naturels. On démontre quelques théorèmes concernant la structure de ces espaces. Entre autre on donne des conditions simples qui impliquent l’existence des unités approchées ou la densité de K(G) dans Λ(A,B,X,G). Dans ce dernier cas on peut caractériser les idéaux fermés.

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     author = {Feichtinger, Hans G.},
     title = {On a class of convolution algebras of functions},
     journal = {Annales de l'Institut Fourier},
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Feichtinger, Hans G. On a class of convolution algebras of functions. Annales de l'Institut Fourier, Volume 27 (1977) no. 3, pp. 135-162. doi : 10.5802/aif.665. http://www.numdam.org/articles/10.5802/aif.665/

[1] B.A. Barnes, Banach algebras which are ideals in a Banach algebra, Pac. J. Math., 38 (1971), 1-7. | MR | Zbl

[2] P.L. Butzer-K. Scherer, Approximationsprozesse und Interpolationsmethoden, Bibl. Inst. Mannheim, 1968. | Zbl

[3] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math., 96 (1956), 1-66. | MR | Zbl

[4] H.G. Feichtinger, Some new subalgebras of L1 (G), Indag. Math., 36 (1974), 44-47. | MR | Zbl

[5] E. Hille, Functional analysis and semigroups, Amer. Math. Soc. Publ., XXXI (1948). | MR | Zbl

[6] A. Hulanicki, On the spectrum of convolution operators on groups of polynomial growth, Invent. math., 17 (1972), 135-142. | MR | Zbl

[7] H. Reiter, Classical harmonic analysis and locally compact groups, Oxford University Press, 1968. | MR | Zbl

[8] R. Spector, Sur la structure locale des groupes abéliens localement compacts, Bull. Soc. Math. France, Mémoire 24 (1970). | Numdam | MR | Zbl

[9] H. Ch. Wang, Nonfactorization in group algebras, Studia math., 42 (1972), 231-241. | MR | Zbl

[10] L.H. Brandenburg, On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl., 50 (1975), 489-510. | MR | Zbl

[11] I.I. Hirschmann, Finite sections of Wiener-Hopf equations and Szegö polynomials, J. Math. Anal. Appl., 11 (1965), 290-320. | MR | Zbl

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