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

Abstract

A functional B on the space of bounded real sequences ℓ∞ is said to be a Banach limit if B⩾0, B(1,1,…)=1 and B(Tx)=B(x) for every x=(x1,x2,…)∈ℓ∞, where T is a translation operator. The set of all Banach limits B is a closed convex set on the unit sphere of ℓ∞⁎. Let C be Cesàro operator, n=1,2,… Denote B(C)={B∈B: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 ℓ1. For given B∈B, r∈(0,2], we denoteSB,r={D∈B:‖D−B‖ℓ=r}. We prove that B∈extB if and only if the sphere SB,r is convex for every r∈(0,2).