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

Abstract

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 ${\ell }^p$, $1\le p<\infty$, 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.

(v1) 3 pp. note, to be submitted (v2) Expanded version, now 7 pp. Extra material includes: more context/motivation: extra example for non-discrete case; new section on the weighted case. Some definitions also clarified