Order and geometric properties of the set of Banach limits

Abstract:

A positive functional $B$ on the space of bounded sequences $\ell _\infty$ is called a Banach limit if $\|B\|_{\ell _\infty ^*}=1$ and $B(x_1, x_2, x_3, \ldots ) = B(0, x_1, x_2, \ldots )$ for every $(x_1, x_2, x_3, \ldots ) \in \ell _\infty$. The set of all Banach limits is denoted by $\mathfrak {B}$ and the set of its extreme points is denoted by $\mathrm {ext}\thinspace \mathfrak {B}$. Various properties of these sets are studied. For instance, there exists $B \in \mathrm {ext}\thinspace \mathfrak {B}$ such that $Bx = 0$ if $x \in \ell _\infty$ and $\lim _{n \to \infty }(|x_1|+\cdots +|x_n|)/n = 0.$ The set $\mathfrak {B}$ fails to possess the $\mathrm {FP}$-property for an affine nonexpansive sequentially weak$^*$ continuous mapping. A general result is proved, which implies that there is a wide class of subspaces of $\ell _\infty$, defined in terms of Banach limits, that are not complemented in $\ell _\infty$. In particular, this class includes the stabilizer $\mathcal {D}(ac_0)$ and the ideal stabilizer $\mathcal {I}(ac_0)$ of the subspace $ac_0$ of almost convergent sequences. In the second part of the paper, the object of study is the set $\mathfrak {B}(\sigma _m)$ of all Banach limits invariant under the dilation operator $\sigma _m$, $m\in {\mathbb N}$ on $\ell _\infty$ given by \[ \sigma _m(x_1,x_2,\ldots )=(\underbrace {x_1,x_1,\ldots ,x_1}_m, \underbrace {x_2,x_2,\ldots ,x_2}_m, \ldots ). \] If $m\geq 2$, then for all $i\in \mathbb {N}$, $i\geq 2$, the inclusion $\mathfrak {B}(\sigma _m)\subseteq \mathfrak {B}(\sigma _{m^i})$ is proper; there exists $B\in \mathfrak {B}(\sigma _m)$ such that $B\notin \mathfrak {B}(\sigma _n)$ for all $n\in F_m=\mathbb {N}\setminus \{1,m,m^2,m^3,\ldots \}$ and $\|B-B_1\|_{l_\infty ^*}=2$ for all $B_1\in \mathfrak {B}(\sigma _n)$ if $n^j\in F_m$ for all $j\in \mathbb {N}$. If $B_1 \in \mathfrak B(\sigma _m)$, $B_2 \in \mathrm {ext}\thinspace \mathfrak B$, then $\|B_1 - B_2\|_{\ell _\infty ^*}=2$. Moreover, the cardinalities of the extreme point are estimated for some subsets of $\mathfrak {B}$. In particular, $\mathrm {card}\thinspace \big (\mathrm {ext}\thinspace \bigcap _{m=1}^\infty \mathfrak {B}(\sigma _m)\big ) = 2^\mathfrak {c}$.