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

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>2010
<mark>Journal</mark>Canadian Mathematical Bulletin
Issue number3
Volume53
Number of pages6
Pages (from-to)447-452
Publication StatusPublished
<mark>Original language</mark>English

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.