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

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

2006-06-15

arxiv.org/abs/math/0606367v2

Canadian Math. Bull. 53 (2010), no. 3, 447--452

Mathematics

Functional Analysis

(v1) 3 pp. note, to be submitted (v2) Expanded version, now 7 pp. Extra material includes: more context/motivation: extra exam

Scientific paper

10.4153/CMB-2010-053-5

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.

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