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

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

In: Canadian Mathematical Bulletin, Vol. 53, No. 3, 2010, p. 447-452.

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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

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 3

ER -