Weighted $L_p$-Algebras on Groups

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.