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Russian Mathematical Surveys, 2020, Volume 75, Issue 4, Pages 725–763
DOI: https://doi.org/10.1070/RM9901 (Mi rm9901) 		 
This article is cited in 16 scientific papers (total in 16 papers)

Surveys

Geometry of Banach limits and their applications

E. M. Semenova, F. A. Sukochevb, A. S. Usachevac

a Voronezh State University
b School of Mathematics and Statistics, University of New South Wales, Sydney, Australia
c Central South University, Changsha, China
English version PDF (836 kB) Citations (16) Russian version article
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DOI: https://doi.org/10.1070/RM9901
Abstract: A Banach limit is a positive shift-invariant functional on $\ell_\infty$ which extends the functional
$$ (x_1,x_2,\dots)\mapsto\lim_{n\to\infty}x_n $$
from the set of convergent sequences to $\ell_\infty$. The history of Banach limits has its origins in classical papers by Banach and Mazur. The set of Banach limits has interesting properties which are useful in applications. This survey describes the current state of the theory of Banach limits and of the areas in analysis where they have found applications.
Bibliography: 137 titles.
Keywords: Banach limits, invariant Banach limits, almost convergent sequences, extreme points, Cesàro operator, dilation operator, Stone–Čech compactification, singular trace of an operator, non-commutative geometry.
Funding agency Grant number
Russian Science Foundation 19-11-00197
Australian Research Council
The research of the first and third authors was supported by the Russian Science Foundation under grant no. 19-11-00197. The research of the second author was carried out with the support of the Australian Research Council.
Received: 03.07.2019
Bibliographic databases:
Document Type: Article
UDC: 517.982.22
MSC: Primary 46B20, 46B45; Secondary 46A22, 46L87, 47L20
Language: English
Original paper language: Russian
Citation: E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Geometry of Banach limits and their applications”, Russian Math. Surveys, 75:4 (2020), 725–763
Citation in format AMSBIB
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\paper Geometry of Banach limits and their applications
\jour Russian Math. Surveys
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\vol 75
\issue 4
\pages 725--763
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    • This publication is cited in the following 16 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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