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

Let $\Gamma$ be a discrete group and let $f \in \ell^1(\Gamma)$. We observe that if the natural convolution operator $\rho_f:\ell^{\infty}(\Gamma)\to \ell^{\inf ty}(\Gamma)$ is injective, then f is invertible in $\ell^1(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra $\ell^1(\Gamma)$. 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 $\Gamma$, and give some partial results.