Functional analysis
On the structure of invariant Banach limits
Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1195-1199.

A functional B on the space of bounded real sequences is said to be a Banach limit if B0, B(1,1,)=1 and B(Tx)=B(x) for every x=(x1,x2,), where T is a translation operator. The set of all Banach limits B is a closed convex set on the unit sphere of . Let C be Cesàro operator (Cx)n=1nk=1nxk, n=1,2, Denote B(C)={BB:B=BC}.

The cardinality of the set of extreme points extB(C) is 2c, where c is the cardinality of continuum. A subspace generated by any countable collection from extB(C) is isometric to 1. For given BB, r(0,2], we denote

SB,r={DB:DB=r}.
We prove that BextB if and only if the sphere SB,r is convex for every r(0,2).

Une forme linéaire B sur l'espace des suites bornées est appelée une limite de Banach si B0, B(1,1,)=1 et B(Tx)=B(x) pour tout x=(x1,x2,), T désignant l'opérateur de translation. L'ensemble B des limites de Banach est un sous-ensemble convexe fermé de la shère unité de . Soit C l'opérateur de Cesàro, (Cx)n=1nk=1nxk, n=1,2, Posons B(C)={BB:B=BC}.

La cardinalité de l'ensemble des points extrémaux extB(C) est 2c, où c désigne la cardinalité du continuum. Un sous-espace engendré par une famille dénombrable de extB(C) est isométrique à 1. Étant donnés BB et r(0,2], notons

SB,r={DB:DB=r}.
Nous montrons que BextB si et seulement si la sphère SB,r est convexe pour tout r(0,2).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.10.007
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
@article{CRMATH_2016__354_12_1195_0,
     author = {Alekhno, Egor and Semenov, Evgeniy and Sukochev, Fedor and Usachev, Alexandr},
     title = {On the structure of invariant {Banach} limits},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1195--1199},
     publisher = {Elsevier},
     volume = {354},
     number = {12},
     year = {2016},
     doi = {10.1016/j.crma.2016.10.007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.10.007/}
}
TY  - JOUR
AU  - Alekhno, Egor
AU  - Semenov, Evgeniy
AU  - Sukochev, Fedor
AU  - Usachev, Alexandr
TI  - On the structure of invariant Banach limits
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1195
EP  - 1199
VL  - 354
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.10.007/
DO  - 10.1016/j.crma.2016.10.007
LA  - en
ID  - CRMATH_2016__354_12_1195_0
ER  - 
%0 Journal Article
%A Alekhno, Egor
%A Semenov, Evgeniy
%A Sukochev, Fedor
%A Usachev, Alexandr
%T On the structure of invariant Banach limits
%J Comptes Rendus. Mathématique
%D 2016
%P 1195-1199
%V 354
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.10.007/
%R 10.1016/j.crma.2016.10.007
%G en
%F CRMATH_2016__354_12_1195_0
Alekhno, Egor; Semenov, Evgeniy; Sukochev, Fedor; Usachev, Alexandr. On the structure of invariant Banach limits. Comptes Rendus. Mathématique, Volume 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/

[1] Alekhno, E.; Semenov, E.; Sukochev, F.; Usachev, A. Order and geometric properties of the set of Banach limits, Algebra Anal., Volume 28 (2016) no. 3, pp. 3-35

[2] Aliprantis, C.D.; Border, K.C. Infinite Dimensional Analysis. A Hitchhiker's Guide, Springer, Berlin, 2006

[3] Banach, S. Théorie des Opérations Linéaires, Éditions Jacques Gabay, Sceaux, France, 1993 (reprint of the 1932 original)

[4] Carey, A.; Phillips, J.; Sukochev, F. Spectral flow and Dixmier traces, Adv. Math., Volume 173 (2003) no. 1, pp. 68-113

[5] Chou, C. On the size of the set of left invariant means on a semi-group, Proc. Amer. Math. Soc., Volume 23 (1969), pp. 199-205

[6] Dodds, P.G.; de Pagter, B.; Sedaev, A.A.; Semenov, E.M.; Sukochev, F.A. Singular symmetric functionals and Banach limits with additional invariance properties, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 67 (2003) no. 6, pp. 111-136

[7] Eberlein, W.F. Banach–Hausdorff limits, Proc. Amer. Math. Soc., Volume 1 (1950), pp. 662-665

[8] Lorentz, G.G. A contribution to the theory of divergent sequences, Acta Math., Volume 80 (1948), pp. 167-190

[9] Meyer-Nieberg, P. Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991

[10] Semenov, E.M.; Sukochev, F.A. Invariant Banach limits and applications, J. Funct. Anal., Volume 259 (2010) no. 6, pp. 1517-1541

[11] Semenov, E.M.; Sukochev, F.A. Extreme points of the set of Banach limits, Positivity, Volume 17 (2013) no. 1, pp. 163-170

[12] Semenov, E.M.; Sukochev, F.A.; Usachev, A.S. Geometric properties of the set of Banach limits, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 78 (2014) no. 3, pp. 177-204

[13] Sucheston, L. Banach limits, Amer. Math. Monthly, Volume 74 (1967), pp. 308-311

Cited by Sources: