Invariant Banach limits and their extreme points
Volume 242 / 2018
Abstract
The set $\mathfrak B $ of all Banach limits is the set of all positive, normalised and shift-invariant functionals on the space $\ell _\infty $ of all bounded sequences. Motivated by W. Eberlein’s work we introduce and study the class of operators $W$ (we call them (strongly) $\mathfrak {B}$-regular) which have at least one $W$-invariant Banach limit. This class includes such well-known operators on $\ell _\infty $ as (generalised) Cesàro and dilation operators. We study the geometric properties of the set of all $W$-invariant Banach limits, and show that the extreme points of this set are multiplicative on the set ${\rm St}(W)$. This is an extension of the statement about the stabiliser for the shift operator introduced by W. Luxemburg. We show that the cardinality of the set of extreme points of the set of Cesàro invariant Banach limits coincides with that of the set of all bounded linear functionals on $\ell _\infty $ and equals $2^\mathfrak {c}$.
