Distance formulas in group algebras

Heybetkulu Mustafayev

doi:10.1016/j.crma.2016.04.002

Mathematical analysis/Functional analysis

Distance formulas in group algebras

Presented by: the Editorial Board

Heybetkulu Mustafayev ¹

¹ Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, 65080, Van, Turkey

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 577-582.

Abstract
Résumé

Let G be a locally compact amenable group, $A (G)$ and $B (G)$ be the Fourier and the Fourier–Stieltjes algebra of G, respectively. For a given $u \in B (G)$ , let $E_{u} : = {g \in G : | u (g) | = 1}$ . The main result of this paper particularly states that if ${‖ u ‖}_{B (G)} \leq 1$ and $\overline{u (E_{u})}$ is countable (in particular, if $E_{u}$ is compact and scattered), then

\lim_{n \to \infty} {‖ u^{n} v ‖}_{A (G)} = dist (v, I_{E_{u}}), \forall v \in A (G),

where

I_{E_{u}} = {v \in A (G) : v (g) = 0, \forall g \in E_{u}}

.

Soit G un groupe compact moyennable et soient $A (G)$ et $B (G)$ l'algèbre de Fourier et l'algèbre de Fourier–Stieltjes de G, respectivement. Pour un $u \in B (G)$ donné, posons $E_{u} : = {g \in G : | u (g) | = 1}$ . Le résultat principal de cet article établit que, si ${‖ u ‖}_{B (G)} \leq 1$ et si $\overline{u (E_{u})}$ est dénombrable (en particulier si $E_{u}$ est compacte et éparpillé), alors

\lim_{n \to \infty} {‖ u^{n} v ‖}_{A (G)} = dist (v, I_{E_{u}}), \forall v \in A (G),

où

I_{E_{u}} = {v \in A (G) : v (g) = 0, \forall g \in E_{u}}

.

Received: 2016-02-13
Accepted: 2016-04-05
Published online: 2016-04-16

DOI: 10.1016/j.crma.2016.04.002

Author's affiliations:

Heybetkulu Mustafayev ¹

¹ Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, 65080, Van, Turkey

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     author = {Heybetkulu Mustafayev},
     title = {Distance formulas in group algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {577--582},
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     doi = {10.1016/j.crma.2016.04.002},
     language = {en},
}

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Heybetkulu Mustafayev. Distance formulas in group algebras. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 577-582. doi : 10.1016/j.crma.2016.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.04.002/

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