Abstract view
Injective Convolution Operators on l^{∞}(Γ) are Surjective


Published:20100511
Printed: Sep 2010
Yemon Choi,
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, Canada
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.
