Convolution-Dominated Operators on Discrete Groups
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- Fendler, G., Gröchenig, K. & Leinert, M. Integr. equ. oper. theory (2008) 61: 493. doi:10.1007/s00020-008-1604-7
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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 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 ℓ2(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 L1-algebras and the symmetry of group algebras.