Abstract: We study the convolution operators which are tauberian as operators acting on the group algebras , where G is a locally compact abelian group and ? is a complex Borel measure on G. We show that these operators are invertible when G is non-compact, and that they are Fredholm when they have closed range and G is compact. In the remaining case, when G is compact and is not assumed to be closed, we prove that is Fredholm when the singular continuous part of ? with respect to the Haar measure on G is zero.

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