On the structure of invariant Banach limits

Alekhno, Egor; Semenov, Evgeniy; Sukochev, Fedor; Usachev, Alexandr

doi:10.1016/j.crma.2016.10.007

Functional analysis

On the structure of invariant Banach limits
[Sur la structure des limites de Banach invariantes]

Présenté par : Gilles Pisier

Alekhno, Egor ¹ ; Semenov, Evgeniy ² ; Sukochev, Fedor ³ ; Usachev, Alexandr ³

¹ Belarusian State University, pr. Nezavisimosti 4, Minsk, 220030, Belarus
² Mathematical Faculty, Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia
³ School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia

Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1195-1199.

Résumé
Abstract

Une forme linéaire B sur l'espace $ℓ_{\infty}$ des suites bornées est appelée une limite de Banach si $B ⩾ 0$ , $B (1, 1, \dots) = 1$ et $B (T x) = B (x)$ pour tout $x = (x_{1}, x_{2}, \dots) \in ℓ_{\infty}$ , T désignant l'opérateur de translation. L'ensemble $B$ des limites de Banach est un sous-ensemble convexe fermé de la shère unité de $ℓ_{\infty}^{⁎}$ . Soit C l'opérateur de Cesàro, ${(C x)}_{n} = \frac{1}{n} \sum_{k = 1}^{n} x_{k}$ , $n = 1, 2, \dots$ Posons $B (C) = {B \in B : B = B C}$ .

La cardinalité de l'ensemble des points extrémaux $ext B (C)$ est $2^{c}$ , où c désigne la cardinalité du continuum. Un sous-espace engendré par une famille dénombrable de $ext B (C)$ est isométrique à $ℓ_{1}$ . Étant donnés $B \in B$ et $r \in (0, 2]$ , notons

S_{B, r} = {D \in B : {‖ D - B ‖}_{ℓ_{\infty}^{⁎}} = r} .

Nous montrons que

B \in ext B

si et seulement si la sphère

S_{B, r}

est convexe pour tout

r \in (0, 2)

A functional B on the space of bounded real sequences $ℓ_{\infty}$ is said to be a Banach limit if $B ⩾ 0$ , $B (1, 1, \dots) = 1$ and $B (T x) = B (x)$ for every $x = (x_{1}, x_{2}, \dots) \in ℓ_{\infty}$ , where T is a translation operator. The set of all Banach limits $B$ is a closed convex set on the unit sphere of $ℓ_{\infty}^{⁎}$ . Let C be Cesàro operator ${(C x)}_{n} = \frac{1}{n} \sum_{k = 1}^{n} x_{k}$ , $n = 1, 2, \dots$ Denote $B (C) = {B \in B : B = B C}$ .

The cardinality of the set of extreme points $ext B (C)$ is $2^{c}$ , where c is the cardinality of continuum. A subspace generated by any countable collection from $ext B (C)$ is isometric to $ℓ_{1}$ . For given $B \in B$ , $r \in (0, 2]$ , we denote

S_{B, r} = {D \in B : {‖ D - B ‖}_{ℓ_{\infty}^{⁎}} = r} .

We prove that

B \in ext B

if and only if the sphere

S_{B, r}

is convex for every

r \in (0, 2)

Reçu le : 2016-06-11
Accepté le : 2016-10-10
Publié le : 2016-10-18

DOI : 10.1016/j.crma.2016.10.007

Affiliations des auteurs :

Alekhno, Egor ¹ ; Semenov, Evgeniy ² ; Sukochev, Fedor ³ ; Usachev, Alexandr ³

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     title = {On the structure of invariant {Banach} limits},
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Alekhno, Egor; Semenov, Evgeniy; Sukochev, Fedor; Usachev, Alexandr. On the structure of invariant Banach limits. Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1195-1199. doi : 10.1016/j.crma.2016.10.007. http://www.numdam.org/articles/10.1016/j.crma.2016.10.007/

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