, Volume 61, Issue 4, pp 493-509
Date: 25 Jul 2008

Convolution-Dominated Operators on Discrete Groups

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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 xG 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 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.