# zbMATH — the first resource for mathematics

Injective convolution operators on $$\ell^\infty(\Gamma)$$ are surjective. (English) Zbl 1211.43001
Summary: Let $$\Gamma$$ be a discrete group and let $$f \in \ell^1(\Gamma)$$. We observe that if the natural convolution operator $$\rho_f: \ell^\infty(\Gamma)\rightarrow \ell^\infty(\Gamma)$$ 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:
 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.
Full Text: