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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
DOI: https://doi.org/10.1090/S0002-9939-98-04237-3
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.


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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
Email: huang@mipool.uni-jena.de

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
Email: J.vanNeerven@twi.tudelft.nl

Frank Räbiger
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, F. R. Germany
Email: frra@michelangelo.mathematik.uni-tuebingen.de

DOI: https://doi.org/10.1090/S0002-9939-98-04237-3
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