On Banach–Mazur limits

A positive functional x∗x∗ on the space ℓ∞ℓ∞ of all bounded sequences is called a  Banach–Mazur limit   if ‖x∗‖=1‖x∗‖=1 and x∗x=x∗Txx∗x=x∗Tx for all x=(x1,x2,…)∈ℓ∞x=(x1,x2,…)∈ℓ∞, where TT is the forward shift operator on ℓ∞ℓ∞, i.e., Tx=(0,x1,x2,…)Tx=(0,x1,x2,…). The set of all Banach–Mazur limits is denoted by BM and a collection of extreme points of BM is denoted by extBM. Let ac0={x∈ℓ∞:x∗x=0for allx∗∈BM}. The following sequence spaces D(ac0)={x∈ℓ∞:x⋅ac0⊆ac0}andI(ac0)=ac0+−ac0+ are studied. In particular, if z∈ℓ∞z∈ℓ∞ then z∈D(ac0)z∈D(ac0) iff z−Tz∈I(ac0)z−Tz∈I(ac0); moreover, z∈D(ac0)z∈D(ac0) iff x∗{n:|zn−x∗z|≥ϵ}=0x∗{n:|zn−x∗z|≥ϵ}=0 for all ϵ>0ϵ>0 and x∗∈extBM. Order properties of Banach–Mazur limits are considered. Some properties of extBM are derived. We used the representation of functionals x∗∈BM as Borel measures on βN∖NβN∖N. The cardinalities of some subset of BM are given. We also consider some questions of the probability theory for finite additive measures. E.g., for every x∗∈BM there exists an element x∈ℓ∞x∈ℓ∞ such that the distribution function Fx∗,x(t)=x∗{n:xn≤t}Fx∗,x(t)=x∗{n:xn≤t} is continuous on  RR. Two definitions of a variance are suggested. It is shown that Radon–Nikodym theorem is not valid for finite additive measures: the relations 0≤x∗≤y∗∈ℓ∞∗ do not imply the existence of w∈ℓ∞w∈ℓ∞ satisfying x∗x=y∗(wx)x∗x=y∗(wx) for all x∈ℓ∞x∈ℓ∞.