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**Injective convolution operators on $\ell^infty(\Gamma)$ are surjective.** / Choi, Yemon.

Research output: Contribution to journal › Journal article › peer-review

Choi, Y 2010, 'Injective convolution operators on $\ell^infty(\Gamma)$ are surjective', *Canadian Mathematical Bulletin*, vol. 53, no. 3, pp. 447-452. https://doi.org/10.4153/CMB-2010-053-5

Choi, Y. (2010). Injective convolution operators on $\ell^infty(\Gamma)$ are surjective. *Canadian Mathematical Bulletin*, *53*(3), 447-452. https://doi.org/10.4153/CMB-2010-053-5

Choi Y. Injective convolution operators on $\ell^infty(\Gamma)$ are surjective. Canadian Mathematical Bulletin. 2010;53(3):447-452. https://doi.org/10.4153/CMB-2010-053-5

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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",

}

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

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 3

ER -