directly finite, Dedekind finite, convolution operators, weighted groups
Appeared as Canadian Math. Bull. 53 (2010), no.3, 447-452.
Preprint version available at arXiv math.FA/0606367
[ Math Review | Zentralblatt (summary)]
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.
This is work from 2006/07, and has been rather overtaken by the sequel paper in IEOT.
The aforementioned paper of Deninger and Schmidt has appeared as
C. Deninger, K. Schmidt. Expansive algebraic actions of discrete residually finite amenable groups and their entropy, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 769--786. [Math Reviews| Zbl ]