Injective convolution operators on ${\ell}^{\infty}(Γ)$ are surjective

Abstract

Let $Γ$ be a discrete group and let $f \in \ell^1(Γ)$. We observe that if the natural convolution operator $ρ_f:\ell^{\infty}(Γ)\to \ell^{\inf ty}(Γ)$ is injective, then f is invertible in $\ell^1(Γ)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra $\ell^1(Γ)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^p$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $Γ$, and give some partial results.