Home > Research > Publications & Outputs > Injective convolution operators on $\ell^infty(... ### Associated organisational units ### Links ### Text available via DOI: ## Injective convolution operators on$\ell^infty(\Gamma)$are surjective Research output: Contribution to journalJournal articlepeer-review Published ### Standard In: Canadian Mathematical Bulletin, Vol. 53, No. 3, 2010, p. 447-452. Research output: Contribution to journalJournal articlepeer-review ### Harvard ### APA ### Vancouver ### Author Choi, Yemon. / Injective convolution operators on$\ell^infty(\Gamma)$are surjective. In: Canadian Mathematical Bulletin. 2010 ; Vol. 53, No. 3. pp. 447-452. ### Bibtex @article{5b6e21bc77404a9e82e05be6bc9b97fc, title = "Injective convolution operators on$\ell^infty(\Gamma)$are surjective", abstract = "Let Γ be a discrete group and let f ∈ l1(Γ). We observe that if the natural convolution operator ρf:l∞(Γ)→ l∞(Γ) is injective, then f is invertible in l1(Γ). Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra l1(Γ). We give simple examples to show that in general one cannot replace l∞ with lp, 1≤ p < ∞, nor with L∞(G) for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on Γ, and give some partial results.", author = "Yemon Choi", year = "2010", doi = "10.4153/CMB-2010-053-5", language = "English", volume = "53", pages = "447--452", journal = "Canadian Mathematical Bulletin", issn = "0008-4395", publisher = "Canadian Mathematical Society", number = "3", } ### RIS TY - JOUR T1 - Injective convolution operators on$\ell^infty(\Gamma)\$ are surjective

AU - Choi, Yemon

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

U2 - 10.4153/CMB-2010-053-5

DO - 10.4153/CMB-2010-053-5

M3 - Journal article

VL - 53

SP - 447

EP - 452