Abstract: We study the convolution operators T?acting on the group algebras L1(G)and M(G), where Gis a locally compact abelian group and ?is a complex Borel measure on G. We show that a cotauberian convolution operator T?acting on L1(G)is Fredholm of index zero, and that T?is tauberian if and only if so is the corresponding convolution operator acting on the algebra of measures M(G), and we give some applications of these results.

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