Convolution operators on group algebras which are tauberian or cotauberian

We study the convolution operators Tμ acting on the group algebras L1(G) and M(G), where G is a locally compact abelian group and μ is a complex Borel measure on G. We show that a cotauberian convolution operator Tμ acting on L1(G) is Fredholm of index zero, and that Tμ 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.