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Order and geometric properties of the set of Banach limits

Authors: E. Alekhno, E. Semenov, F. Sukochev and A. Usachev
Translated by: the authors
Original publication: Algebra i Analiz, tom 28 (2016), nomer 3.
Journal: St. Petersburg Math. J. 28 (2017), 299-321
MSC (2010): Primary 46B15
Published electronically: March 29, 2017
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Abstract: A positive functional $ B$ on the space of bounded sequences $ \ell _\infty $ is called a Banach limit if $ \Vert B\Vert _{\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 }(\vert x_1\vert+\cdots +\vert x_n\vert)/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

$\displaystyle \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 $ \Vert B-B_1\Vert _{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 $ \Vert B_1 - B_2\Vert _{\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}$.

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Additional Information

E. Alekhno
Affiliation: Department of Mechanics and Mathematics, Belorussian State University, pr. Nezavisimosti 4, 220030 Minsk, Belorussia

E. Semenov
Affiliation: Department of Mathematics, Voronezh State University, Universitetskaya pl. 1, 394006 Voronezh, Russia

F. Sukochev
Affiliation: School of Mathematics and Statistics, New South Wales University, Kensington 2052, New South Wales Australia

A. Usachev
Affiliation: School of Mathematics and Statistics, New South Wales University, Kensington 2052, New South Wales Australia

Keywords: Banach limit, space of bounded sequences $\ell_\infty$, dilation operator
Received by editor(s): December 25, 2015
Published electronically: March 29, 2017
Additional Notes: The second author was supported by RFBR (grant no. 14-01-00141a). The third and fourth authors were partially supported by the Australian Research Council (grant DP140100906).
Article copyright: © Copyright 2017 American Mathematical Society