Convolution-dominated operators on discrete groups

摘要

We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that vertical bar(Ac)(x)vertical bar <= (a*vertical bar c vertical bar)(x) for x is an element of G and some a is an element of l(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.