Convolution-Dominated Operators on Discrete Groups

We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that for x ∈ G and some . This class of “convolution-dominated” matrices forms a Banach-*-algebra contained in the algebra of bounded operators on ℓ ²(G). Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that G is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized L ¹-algebras and the symmetry of group algebras. K. G. was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154.