Norm-controlled inversion in weighted convolution algebras

TL;DR

This paper establishes conditions under which weighted convolution algebras on discrete groups have a norm-controlled inversion property within their reduced C*-algebras, extending to non-discrete cases and various group types.

Contribution

It provides new sufficient conditions for norm-controlled inversion in weighted convolution algebras on discrete groups, including non-discrete cases and specific group classes.

Findings

Applicable to locally finite groups and finitely generated groups of polynomial or intermediate growth.

Includes weights of polynomial or subexponential type.

Extends results to non-discrete groups and related operator algebras.

Abstract

Let $G$ be a discrete group, let $p\ge1$, and let $\omega$ be a weight on $G$. Using the approach from [9], we provide sufficient conditions on a weight $\omega$ for $\ell^p(G,\omega)$ to be a Banach algebra admitting a norm-controlled inversion in the reduced C$^*$-algebra of $G$, namely $C^*_r(G)$. We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in $B(L^2(G))$ for some related convolution algebras.