On Banach-Mazur limits

Abstract

A positive functional x* on the space ℓ∞ of all bounded sequences is called a Banach-Mazur limit if ||x*|| = 1 and x*x = x*Tx for all x = (x1, x2,..)∈ ℓ∞, where T is the forward shift operator on ℓ∞, i.e., 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 ext BM. Let ac0 = {x ∈ ℓ∞: x*x = 0 for all x* ∈ BM}. The following sequence spaces D(ac0) = {x ∈ ℓ∞: x ·ac0 ⊆ ac0} and I (ac0) = ac0+-ac0+ are studied. In particular, if z ∈ ℓ∞ then z∈ D(ac0) iff z-Tz∈ I(ac0); moreover, z∈D(ac0) iff x*{n :|zn-x*z|≥ε} = 0 for all ε > 0 and x*∈ext BM. Order properties of Banach-Mazur limits are considered. Some properties of ext BM are derived. We used the representation of functionals x*∈BM as Borel measures on βN\N. Thecardinalities 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 anelement x∈ℓ∞ such that thedistribution function Fx*,x(t)= x*{n: xn ≤t} is continuous on ℝ. Two definitions of avariance 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 theexistence of w ∈ ℓ∞ satisfying x* x = y*(wx) for all x ∈ℓ∞.