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 Gröchenig and Klotz (J Lond Math Soc (2) 88:49–64, 2013), we provide sufficient conditions on a weight $$\omega $$ for $$\ell ^p(G,\omega )$$ to be a Banach algebra admitting a norm-controlled inversion in $$C^*_r(G)$$. 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.