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# Convergence Factors and Compactness in Weighted Convolution Algebras

Published:2002-04-01
Printed: Apr 2002
• Fereidoun Ghahramani
• Sandy Grabiner
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## Abstract

We study convergence in weighted convolution algebras $L^1(\omega)$ on $R^+$, with the weights chosen such that the corresponding weighted space $M(\omega)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta$ for which weak$^\ast$-convergence of $\{\lambda_n\}$ to $\lambda$ in $M(\omega)$ implies norm convergence of $\lambda_n \ast f$ to $\lambda \ast f$ in $L^1 (\omega\eta)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all $L^1(\omega)$ and all $f$ in $L^1(\omega)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M(\omega)$ (or $L^1(\omega)$) to $L^1 (\omega\eta)$.
 MSC Classifications: 43A10 - Measure algebras on groups, semigroups, etc. 43A15 - $L^p$-spaces and other function spaces on groups, semigroups, etc. 46J45 - Radical Banach algebras 46J99 - None of the above, but in this section

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