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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.

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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