Convolution-Dominated Operators on Discrete Groups

TL;DR

This paper investigates convolution-dominated matrices on discrete groups, proving their algebraic inverse-closedness under certain conditions, extending classical results from abelian to non-abelian groups using advanced algebraic techniques.

Contribution

It establishes that convolution-dominated matrices form an inverse-closed Banach algebra on amenable, rigidly symmetric groups, generalizing known results to non-abelian groups.

Findings

Convolution-dominated matrices form a Banach-*-algebra.

Inverses of such matrices are also convolution-dominated.

The results extend classical abelian group cases to non-abelian 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 $x\in G$ 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 $\ell ^2(G)$. 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.