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

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