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Ditkin's condition for certain Beurling algebras

Authors: Sen-Zhong Huang, Jan van Neerven and Frank Räbiger
Journal: Proc. Amer. Math. Soc. 126 (1998), 1397-1407
MSC (1991): Primary 43A45, 43A20, 47D03
MathSciNet review: 1443833
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Abstract: Let $G$ be a locally compact abelian group. A function $\omega:G\to[1,\infty)$ is said to be a weight if it is locally bounded, Borel measurable and submultiplicative. We call a weight $\omega$ on $G$ semi-bounded if there exist a constant $K$ and a subsemigroup $S$ with $S-S=G,$ such that

\begin{displaymath}\omega(s)\leq K\quad \text{and}\quad \lim _{n\to\infty}\frac{\log\omega(-ns)}{\sqrt{n}}=0\end{displaymath}

for all $s\in S.$ Using functional analytic methods, we show that all Beurling algebras $L^1_\omega(G)$ whose defining weight $\omega$ is semi-bounded satisfy Ditkin's condition.

References [Enhancements On Off] (What's this?)

  • 1. W. ARVESON: On groups of automorphisms of operator algebras. J. Funct. Anal. 15 (1974), 217-243. MR 50:1016
  • 2. A. ATZMON: Operators which are annihilated by analytic functions. Acta Math. 144 (1980), 27-63. MR 81c:47007
  • 3. J. J. BENEDETTO: ``Spectral Synthesis,'' B.G. Teubner, Stuttgart (1975). MR 58:29850a
  • 4. A. BEURLING: Sur les intégrales de Fourier absolument convergentes, IX$^e$ Congrès Math. Scand., Helsinki, pp. 345-366 (1966).
  • 5. W. G. BADE AND H. G. DALES: Uniqueness of complete norms for quotients of Banach function algebras. Studia Math. 106 (1993), 289-302. MR 94f:46062
  • 6. C. D'ANTONI, C. LONGO AND L. ZSIDÓ: A spectral mapping theorem for locally compact groups of operators. Pacific J. Math. 103 (1982), 17-24. MR 84e:47058
  • 7. M. M. DAY: ``Normed Linear Spaces,'' 3rd ed., Springer-Verlag, Berlin- Heidelberg-New York (1973). MR 49:9588
  • 8. V. DITKIN: On the structure of ideals in certain normed rings. Uchen. Zap. Mosk. Gos. Univ. Matem. 30 (1939), 81-130. (Russian; English summary) MR 1:336b
  • 9. Y. DOMAR: Harmonic analysis based on certain commutative Banach algebras. Acta Math. 96 (1956), 1-66. MR 17:1228a
  • 10. E. HEWITT AND R. ROSS: ``Abstract Harmonic Analysis I,'' Springer-Verlag, Berlin-Heidelberg-New York (1963). MR 28:158
  • 11. S.-Z. HUANG: ``Spectral Theory for Non-Quasianalytic Representations of Locally Compact Abelian Groups,'' Thesis, Universität Tübingen (1996). A summary has appeard in ``Dissertation Summaries in Mathematics'' 1 (1996), 171-178.
  • 12. Y. KATZNELSON: ``An Introduction to Harmonic Analysis,'' 2nd ed., Dover Publications, New York (1976). MR 54:10976
  • 13. R. LARSEN, T. S. LIU AND J. K. WANG: On functions with Fourier transforms in $L_p.$ Michigan Math. J. 11 (1964), 369-378. MR 30:412
  • 14. L. H. LOOMIS: ``An Introduction to Abstract Harmonic Analysis,'' van Nostrand, New York (1953). MR 14:883c
  • 15. P. MALLIAVIN: Impossibilité de la synthèse spectrale sur les groupes abéliens. Inst. Hautes Études Sci. Publ. Math. 2 (1959), 61-68. MR 21:5854c
  • 16. R. REITER: ``Classical Harmonic Analysis and Locally Compact Groups,'' Oxford Univ. Press, London (1968). MR 46:5933
  • 17. R. REITER: ``$L^1-$Algebras and Segal Algebras,'' Springer-Verlag, Berlin-Heidelberg-New York (1971). MR 55:13158
  • 18. I. E. SEGAL: The group algebra of a locally compact group. Trans. Amer. Math. Soc. 61 (1947), 69-105. MR 8:438c
  • 19. M. WOLFF: Spectral theory of group representations and their nonstandard hull. Israel J. Math. 48 (1984), 205-224. MR 86e:46046
  • 20. M. ZARRABI: Ensembles de synthèse pour certaines algèbres de Beurling. Rev. Roumaine Math. Pures Appl. 35 (1990), 385-396. MR 92a:43005

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Additional Information

Sen-Zhong Huang
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, F. R. Germany
Address at time of publication: Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 1-4, D-07743 Jena, Germany

Jan van Neerven
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, F. R. Germany
Address at time of publication: Department of Mathematics, Tu Delft, P. O. Box 356, 2600 AJ Delft, the Netherlands

Frank Räbiger
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, F. R. Germany

Keywords: Ditkin's condition, group representation, spectrum
Received by editor(s): October 14, 1996
Additional Notes: This research is supported by Deutscher Akademischer Austauschdienst DAAD (first author) and by the Human Capital Mobility Programme of the European Community (second author). It is part of a research project supported by Deutsche Forschungsgemeinschaft DFG (third author).
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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