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Injective Convolution Operators on l(Γ) are Surjective

  Published:2010-05-11
 Printed: Sep 2010
  • Yemon Choi,
    Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, Canada
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Abstract

Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ 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.
MSC Classifications: 43A20, 46L05, 43A22 show english descriptions $L^1$-algebras on groups, semigroups, etc.
General theory of $C^*$-algebras
Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A20 - $L^1$-algebras on groups, semigroups, etc.
46L05 - General theory of $C^*$-algebras
43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
 

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