Norm-controlled inversion in weighted convolution algebras

Abstract

Let $G$ be a discrete group, let $p\ge 1$, 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${}^{\ast }$-algebra of $G$, namely $C_r^{\ast }(G)$. 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.