Injective convolution operators on $\ell^infty(\Gamma)$ are surjective

Choi, Yemon (2010) Injective convolution operators on $\ell^infty(\Gamma)$ are surjective. Canadian Mathematical Bulletin, 53 (3). pp. 447-452. ISSN 0008-4395

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.