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Brief communications

Weighted $L_p$-Algebras on Groups

Yu. N. Kuznetsova

Moscow State Institute of Electronics and Mathematics

Abstract: The space $L_p(G)$, $1<p<\infty$, on a locally compact group $G$ is known to be closed under convolution only if $G$ is compact. However, the weighted spaces $L_p(G,w)$ are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra $L_p(G,w)$ on an Abelian group $G$ is described. It is shown that all algebras of this type are semisimple.

Keywords: weighted convolution algebra, Beurling algebra, multiplicative spectrum, locally compact Abelian group.

UDC: 517.982

Received: 03.06.2005

DOI: 10.4213/faa748

 English version: Functional Analysis and Its Applications, 2006, 40:3, 234–236

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