Injective convolution operators on l∞(Γ) are surjective

Yemon Choi

Keywords

directly finite, Dedekind finite, convolution operators, weighted groups

Status

Preprint version available at arXiv math.FA/0606367

Reviews

[ Math Review | Zentralblatt (summary)]

Abstract

Let Γ be a discrete group and let f ∈ l1(Γ). We observe that if the natural convolution operator ρf:l(Γ)→ l(Γ) is injective, then f is invertible in l1(Γ). Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra l1(Γ).

We give simple examples to show that in general one cannot replace l with lp, 1≤ p < ∞, nor with L(G) for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on Γ, and give some partial results.