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Published online by Cambridge University Press: 20 November 2018
Let $\Gamma $ be a discrete group and let
$f\,\in \,{{l}^{1}}(\Gamma )$ . We observe that if the natural convolution operator
$\rho f\,:\,{{l}^{\infty }}(\Gamma )\,\to \,{{l}^{\infty }}(\Gamma )$ is injective, then
$f$ is invertible in
${{l}^{1}}(\Gamma )$ . Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra
${{l}^{1}}(\Gamma )$ .
We give simple examples to show that in general one cannot replace ${{l}^{\infty }}$ with
${{l}^{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.