Let Γ be a discrete group and let f ∈ ℓ1(Γ). We observe that if the natural convolution operator ρ f : ℓ∞(Γ) → ℓ∞(Γ) is injective, then f is invertible in ℓ1(Γ). Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra ℓ1(Γ).
We give simple examples to show that in general one cannot replace ℓ∞ with ℓ p , 1 ≤ p < ∞, nor with L ∞(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.
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