Convolution-Dominated Operators on Discrete Groups

Abstract.

We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that $|(Ac)(x)| \leq (a \ast |c|)(x)$ for x ∈ G and some $a \in \ell^1(G)$. 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.